cotangent

　 　

WordNet

1. ratio of the adjacent to the opposite side of a right-angled triangle (同)cotan
2. the inverse function of the cotangent; the angle that has a cotangent equal to a given number (同)arccotangent, inverse cotangent

PrepTutorEJDIC

1. コタンジェト,余接

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2020/02/15 10:46:43」(JST)

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三角関数（さんかくかんすう、英: trigonometric function）とは、平面三角法における、角の大きさと線分の長さの関係を記述する関数の族および、それらを拡張して得られる関数の総称である。鋭角を扱う場合、三角関数の値は対応する直角三角形の二辺の長さの比であり、三角関数は三角比とも呼ばれる。三角法に由来する三角関数という呼び名のほかに、後述する単位円を用いた定義に由来する円関数（えんかんすう、英: circular function）という呼び名がある。

三角関数には以下の6つがある。

• sin（正弦、sine）
• sec（正割、secant）
• tan（正接、tangent）
• cos（余弦、cosine）
• csc（余割、cosecant）
• cot（余接、cotangent）

特に sin, cos は幾何学的にも解析学的にも良い性質を持っているので、様々な分野で用いられる。例えば波や電気信号などは正弦関数と余弦関数を組み合わせることで表現することができる。この事実はフーリエ級数およびフーリエ変換の理論として知られ、音声などの信号の合成や解析の手段として利用されている。他にもベクトルの外積や内積は正弦関数および余弦関数を用いて表すことができ、ベクトルを図形に対応づけることができる。初等的には、三角関数は実数を変数とする一変数関数として定義される。三角関数の変数の対応するものとしては、図形のなす角度や、物体の回転角、波や信号のような周期的なものに対する位相などが挙げられる。

三角関数に用いられる独特な記法として、三角関数の累乗と逆関数に関するものがある。通常、関数 f (x) の累乗は (f (x))² = f (x)・f (x) や (f (x))⁻¹ = 1 / f (x) のように書くが、三角関数の累乗は sin²x のように書かれることが多い。逆関数については通常の記法 (f ⁻¹(x)) と同じく、sin⁻¹x などと表す（この文脈では従って、三角関数の逆数は分数を用いて 1/sin x のように、あるいは (sin x)⁻¹ などと表される）。文献あるいは著者によっては、通常の記法と三角関数に対する特殊な記法との混同を避けるため、三角関数の累乗を通常の関数と同様にすることがある。また、三角関数の逆関数として −1 と添え字する代わりに関数の頭に arc とつけることがある（たとえば sin の逆関数として sin⁻¹ の代わりに arcsin を用いる）。

三角関数に似た性質を持つ関数として、指数関数や双曲線関数、ベッセル関数などがある。また、三角関数を利用して定義される関数としてしばしば応用されるものにsinc関数がある。

定義

直角三角形による定義

∠C を直角とする直角三角形ABC

直角三角形において、1 つの鋭角の大きさが決まれば、三角形の内角の和は 180°であることから他の 1 つの鋭角の大きさも決まり、3 辺の比も決まる。ゆえに、角度に対して辺比の値を与える関数を考えることができる。

∠C を直角とする直角三角形 ABC において、それぞれの辺の長さを AB = h, BC = a, CA = b と表す（図を参照）。∠A = θ に対して三角形の辺の比 h : a : b が決まることから、

${\begin{aligned}\sin \theta &={a}/{h}\\\sec \theta &={h}/{b}={1}/{\cos \theta }\\\tan \theta &={a}/{b}={\sin \theta }/{\cos \theta }\\\cos \theta &={b}/{h}\\\operatorname {cosec} \theta &=\csc \theta ={h}/{a}={1}/{\sin \theta }\\\cot \theta &={b}/{a}={1}/{\tan \theta }\end{aligned}}$

という 6 つの値が定まる。それぞれ正弦（sine; サイン）、正割（secant; セカント）、正接（tangent; タンジェント）、余弦（cosine; コサイン）、余割（cosecant; コセカント）、余接（cotangent; コタンジェント）と呼び、まとめて三角比と呼ばれる。ただし cosec は長いので csc と略記することも多い。ある角 ∠A に対する余弦、余割、余接はその角 ∠A の余角 (co-angle) に対する正弦、正割、正接として定義される。

${\begin{aligned}\cos \theta &=\sin \left(90^{\circ }-\theta \right)=\sin(\pi /2-\theta )\\\csc \theta &=\sec \left(90^{\circ }-\theta \right)=\sec(\pi /2-\theta )\\\cot \theta &=\tan \left(90^{\circ }-\theta \right)=\tan(\pi /2-\theta )\end{aligned}}$

三角比は平面三角法に用いられ、巨大な物の大きさや遠方までの距離を計算する際の便利な道具となる。角度 θ の単位は、通常度またはラジアンである。

三角比、すなわち三角関数の直角三角形を用いた定義は、直角三角形の鋭角に対して定義されるため、その定義域は θ が 0° から 90° まで(0 から π / 2 まで)の範囲に限られる。また、θ = 90° (= π / 2) の場合 sec, tan が、θ = 0°(= 0) の場合 csc, cot がそれぞれ定義されない。これは分母となる辺の比の大きさが 0 になるためゼロ除算が発生し、その除算自体が数学的に定義されないからである。一般の角度に対する三角関数を得るためには、三角関数について成り立つ何らかの定理を指針として、定義の拡張を行う必要がある。後述する単位円による定義は初等幾何学におけるそのような拡張の例である。他に同等な方法として、正弦定理や余弦定理を用いる方法などがある。

単位円による定義

単位円による、6つの三角関数が表す長さ

2 次元ユークリッド空間 R² における単位円 {x(t)}² + {y(t)}² = 1 上の点を A = (x(t), y(t)) とする。反時計回りを正の向きとして、原点と円周を結ぶ線分 OA と x 軸のなす角の大きさ ∠xOA を媒介変数 t として選ぶ。このとき実変数 t に対する三角関数は以下のように定義される。

${\begin{aligned}\sin t&=y\\\cos t&=x\\\tan t&={\frac {y}{x}}={\frac {\sin t}{\cos t}}\end{aligned}}$

これらは順に正弦関数 (sine function)、余弦関数 (cosine function)、正接関数(tangent function) と呼ばれる。さらにこれらの逆数として以下の 3 つの関数が定義される。

${\begin{aligned}\csc t&={\frac {1}{y}}={\frac {1}{\sin t}}\\\sec t&={\frac {1}{x}}={\frac {1}{\cos t}}\\\cot t&={\frac {x}{y}}={\frac {1}{\tan t}}\end{aligned}}$

これらは順に余割関数 (cosecant function)、正割関数 (secant function)、余接関数 (cotangent function) と呼ばれ、sin, cos, tan と合わせて三角関数と総称される。特に csc, sec, cot は割三角関数（かつさんかくかんすう）と呼ばれることがある。

この定義は 0 < t < π / 2 の範囲では直角三角形による定義と一致する。

級数による定義

角度、辺の長さといった幾何学的な概念への依存を避けるため、また定義域を複素数に拡張するために、級数を用いて定義することもできる。この定義は実数の範囲では単位円による定義と一致する。以下の級数は共に示される収束円内で収束する。

• z を複素変数、B_n をベルヌーイ数、E_n をオイラー数とする。

${\begin{aligned}\sin z&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}z^{2n+1}\quad {\text{for all}}\ z,\\\cos z&=\sum _{n=0}^{\infty }{\frac {\left(-1\right)^{n}}{\left(2n\right)!}}z^{2n}\quad {\text{for all}}\ z,\\\tan z&=\sum _{n=1}^{\infty }{\frac {\left(-1\right)^{n}2^{2n}\left(1-2^{2n}\right)B_{2n}}{\left(2n\right)!}}z^{2n-1}\quad {\text{for}}\ |z|<{\frac {\pi }{2}},\\\cot z&=\sum _{n=0}^{\infty }{\frac {\left(-1\right)^{n}2^{2n}B_{2n}}{\left(2n\right)!}}z^{2n-1}\quad {\text{for}}\ 0<|z|<\pi ,\\\sec z&=\sum _{n=0}^{\infty }{\frac {\left(-1\right)^{n}E_{2n}}{\left(2n\right)!}}z^{2n}\quad {\text{for}}\ |z|<{\frac {\pi }{2}},\\\csc z&=\sum _{n=0}^{\infty }{\frac {\left(-1\right)^{n}\left(2-2^{2n}\right)B_{2n}}{\left(2n\right)!}}z^{2n-1}\quad {\text{for}}\ 0<|z|<\pi .\end{aligned}}$

微分方程式による定義

実関数 f (x) の二階線型常微分方程式の初期値問題

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   {\displaystyle f''(x)=-f(x),\;f(0)=1,\;f'(0)=0}
 

(1)

の解として cos x を定義し、sin x を −d (cos x)/dx として定義できる。 上記の式を 1 階の連立常微分方程式に書き換えると、g (x) = f ' (x) として、

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   {\displaystyle {\begin{cases}f'(x)=g(x),\\g'(x)=-f(x)\end{cases}}}
 

(2)

および初期条件 f (0) = 1, g (0) = 0 となる。

他の定義

この他にも定積分による（逆三角関数を用いた）定義や複素平面の角の回転による定義などが知られている。

三角関数の性質

周期性

正円より得られる cos θ と sin θ

sin x と cos x のグラフ。これらの関数の周期性が確認できる。

x 軸の正の部分となす角は

$t=\theta +2\pi n\quad (0\leq \theta <2\pi ,\,n\in \mathbb {Z} )$

と表すことができ、θ を偏角、t を一般角と言う。

一般角 t が 2π 進めば点 P(cos t, sin t) は単位円上を1周し元の位置に戻る。従って、

${\begin{aligned}\cos(t+2\pi n)&=\cos t\\\sin(t+2\pi n)&=\sin t\end{aligned}}$

すなわち三角関数 cos, sin は周期 2π の周期関数である。

ほぼ同様に、tan, cot は周期 π の周期関数、sec, csc は周期 2π の周期関数である。

三角関数のグラフ: Sine（青実線）、 Cosine（緑実線）、 Tangent（赤実線）、 Cosecant（青点線）、 Secant（緑点線）、 Cotangent（赤点線）

相互関係

詳細は「三角関数の公式の一覧」を参照

単位円上の点の座標の関数であることから、三角関数の間には多数の相互関係が存在する。

基本相互関係

三角関数の間に成り立つ最も基本的な恒等式の 1 つとして

$\sin ^{2}\theta +\cos ^{2}\theta =1$

が挙げられる。これはピタゴラスの基本三角関数公式 (Fundamental Pythagorean trigonometric identity) と呼ばれている。

上記の式を変形して整理すれば、以下の式が導かれる。

${\begin{aligned}\sec ^{2}\theta -\tan ^{2}\theta &={\frac {1}{\cos ^{2}\theta }}-\tan ^{2}\theta =1,\\\csc ^{2}\theta -\cot ^{2}\theta &={\frac {1}{\sin ^{2}\theta }}-{\frac {1}{\tan ^{2}\theta }}=1.\end{aligned}}$

負角・余角・補角公式

負角

${\begin{aligned}\sin(-\theta )&=-\sin \theta \\\cos(-\theta )&=\cos \theta \\\tan(-\theta )&=-\tan \theta \end{aligned}}$

余角

${\begin{aligned}\sin(\pi /2-\theta )&=\cos \theta \\\cos(\pi /2-\theta )&=\sin \theta \\\tan(\pi /2-\theta )&=\cot \theta \end{aligned}}$

補角

${\begin{aligned}\sin(\pi -\theta )&=\sin \theta \\\cos(\pi -\theta )&=-\cos \theta \\\tan(\pi -\theta )&=-\tan \theta \end{aligned}}$

加法定理

${\begin{aligned}\sin(x\pm y)&=\sin x\cos y\pm \cos x\sin y\\\cos(x\pm y)&=\cos x\cos y\mp \sin x\sin y\\\tan(x\pm y)&={\frac {\tan x\pm \tan y}{1\mp \tan x\tan y}}\end{aligned}}$

余角や補角の公式は加法定理の特別な場合として得られることに注意する。

証明

ピタゴラスの基本三角公式

三角関数および指数関数は冪級数によって定義されているものとすると、負角公式と指数法則およびオイラーの公式より

${\begin{aligned}1&=e^{0}=e^{i\theta -i\theta }=e^{i\theta }e^{-i\theta }\\&=\left(\cos \theta +i\sin \theta \right)\left(\cos \theta -i\sin \theta \right)\\&=\sin ^{2}\theta +\cos ^{2}\theta \end{aligned}}$

である。

負角

sin および cos については、冪級数による表示から明らかである。また

$\tan(-\theta )={\frac {\sin(-\theta )}{\cos(-\theta )}}={\frac {-\sin \theta }{\cos \theta }}=-\tan \theta$

である。

加法定理

オイラーの公式

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   {\displaystyle e^{iz}=\cos z+i\sin z}
 

Euler's formula

と負角の公式から

$\cos z={\frac {e^{iz}+e^{-iz}}{2}},\sin z={\frac {e^{iz}-e^{-iz}}{2i}}$

を得、指数法則

$e^{z+w}=e^{z}e^{w}$

を用いれば sin, cos の加法定理が得られる。これらから他の三角関数についての加法定理も得られる。

PQ（緑の線分の長さ）を求める。

また、三平方の定理から加法定理を示す方法が挙げられる。この方法では、円周上の任意の 2 点間の距離を 2 通りの座標系について求めることで、両者が等しいことから加法定理を導く。2 点間の距離を求めるのに三平方の定理を用いる。以下では単位円のみを取り扱うが、円の半径によらずこの方法から加法定理を得ることができる。

単位円の周上に 2 点 P = (cos p, sin p), Q = (cos q, sin q) を取る。P と Q を結ぶ線分の長さを PQ として、その 2 乗 PQ² を 2 通りの方法で求めることを考える（右図も参照）。

P と Q の x 座標の差と y 座標の差から、三平方の定理を用いて PQ² を求める。

 
   
     
       
         
           
             
               
                 
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   {\displaystyle {\begin{aligned}\mathrm {PQ} ^{2}&=\left(\cos p-\cos q\right)^{2}+\left(\sin p-\sin q\right)^{2}\\&=\left(\cos ^{2}p+\sin ^{2}p\right)+\left(\cos ^{2}q+\sin ^{2}q\right)-2\left(\cos p\cos q+\sin p\sin q\right)\\&=2-2\left(\cos p\cos q+\sin p\sin q\right).\end{aligned}}}
 

(1)

次に Q = (cos 0, sin 0) = (1, 0) となるような座標系を取り、同様に三平方の定理から PQ² を求める。この座標系に対する操作は、x 軸および y 軸を角度 q だけ回転させる操作に相当するので、P = (cos(p − q), sin(p − q)) となる。従って、

 
   
     
       
         
           
             
               
                 
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   {\displaystyle {\begin{aligned}\mathrm {PQ} ^{2}&=\left(\cos(p-q)-1\right)^{2}+\left(\sin(p-q)-0\right)^{2}\\&=2-2\cos(p-q)\end{aligned}}}
 

(2)

となる。

(1) と (2) の右辺が互いに等しいことから、次の cos に関する加法定理が得られる。

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   {\displaystyle {\begin{aligned}\cos p\cos q+\sin p\sin q=\cos(p-q).\end{aligned}}}
 

(3)

三角関数の他の性質を利用することで、(3) から sin の加法定理なども導くことができる。

微積分

三角関数の微積分は、以下の表のとおりである。ただし、これらの結果には様々な（一見同じには見えない）表示が存在し、この表における表示はいくつかの例であることに注意されたい。

$$ f ( x ) {\displaystyle f(x)}	$$ f ′ ( x ) {\displaystyle f'(x)}	$$ ∫ f ( x ) d x {\displaystyle \int f(x)\,dx}
$$ sin ⁡ x {\displaystyle \sin x}	$$ cos ⁡ x {\displaystyle \cos x}	$$ − cos ⁡ x + C {\displaystyle -\cos x+C}
$$ cos ⁡ x {\displaystyle \cos x}	$$ − sin ⁡ x {\displaystyle -\sin x}	$$ sin ⁡ x + C {\displaystyle \sin x+C}
$$ tan ⁡ x {\displaystyle \tan x}	$$ sec 2 ⁡ x = 1 + tan 2 ⁡ x {\displaystyle \sec ^{2}x=1+\tan ^{2}x}	$$ − ln ⁡ | cos ⁡ x | + C {\displaystyle -\ln \left|\cos x\right|+C}
$$ cot ⁡ x {\displaystyle \cot x}	$$ − csc 2 ⁡ x = − ( 1 + cot 2 ⁡ x ) {\displaystyle -\csc ^{2}x=-\left(1+\cot ^{2}x\right)}	$$ ln ⁡ | sin ⁡ x | + C {\displaystyle \ln \left|\sin x\right|+C}
$$ sec ⁡ x {\displaystyle \sec x}	$$ sec ⁡ x tan ⁡ x {\displaystyle \sec x\tan x}	$$ ln ⁡ | sec ⁡ x + tan ⁡ x | + C = gd − 1 ⁡ x + C {\displaystyle \ln \left|\sec x+\tan x\right|+C=\operatorname {gd} ^{-1}x+C}
$$ csc ⁡ x {\displaystyle \csc x}	$$ − csc ⁡ x cot ⁡ x {\displaystyle -\csc x\cot x}	$$ − ln ⁡ | csc ⁡ x + cot ⁡ x | + C = ln ⁡ | tan ⁡ x 2 | + C {\displaystyle -\ln \left|\csc x+\cot x\right|+C=\ln \left|\tan {\frac {x}{2}}\right|+C}

ただし、gd⁻¹x はグーデルマン関数の逆関数である。

三角関数の微分では、次の極限

$\lim _{h\to 0}{\frac {\sin h}{h}}=1$

の成立が基本的である。このとき、sin x の導関数が cos x であることは加法定理から従う（が、後述のようにこれは循環論法であると指摘される）。さらに余角公式 cos x = sin (π /2 − x) から cos x の導関数は −sin x である。すなわち、sin x は微分方程式 y'' (x) + y (x) = 0 の特殊解である。また、他の三角関数の導関数も、上の事実から簡単に導ける。

(sin x)/x の x → 0 における極限

(sin x)/x の x → 0 における極限が 1 であることを証明するときに、中心角 x ラジアンの扇形の面積を2つの三角形の面積ではさんだり、弧長を線分の長さではさんだりして、いわゆるはさみうちの原理から証明する方法がある。これは一般的な日本の高校の教科書にも載っているものであるが、循環論法であるため論理が破綻しているという主張がなされることがある。ここで問題となるのは、証明に面積やラジアン、弧長が利用されていることである。例えば面積について言えば、面積は積分によって定義されるものであるとすると、扇形の面積を求めるには三角関数の積分が必要となる。三角関数の積分をするには三角関数の微分ができなければならないが、三角関数を微分するにはもとの極限が必要になる。このことが循環論法と呼ばれているのである。

単位円板の面積が π であることを自明な概念と考えてしまえば循環論法にはならないが、これはいくつかの決められた公理・定義から論理的演繹のみによって証明されたものだけを正しいと考える現代数学の思想とは相反するものである。循環論法を回避する方法の 1 つは、正弦関数と余弦関数を上述のような無限級数で定義するものである（これは三角関数の標準的な定義の 1 つである。また、この無限級数の収束半径は無限大である（すなわち任意の実数や複素数で収束する））。この定義に基づいて (sin x)/x → 1 (x → 0) を示すことができる。

しかしながら、このように定義された三角関数が、本来持つべき幾何学的な性質を有しているかどうかは全く明らかなことではない。これを確かめるためには、三角関数の諸公式（周期性やピタゴラスの基本三角関数公式等）を証明し、また円周率は、余弦関数の正の最小の零点（つまり、cos x = 0 となる正の最小の値）の存在を示し、その 2 倍と定義する。すると、$$

 
   
     
       x
       ↦
       (
       cos
       ⁡
       x
       ,
       sin
       ⁡
       x
       )
     
   
   {\displaystyle x\mapsto (\cos x,\sin x)}
 

が区間 [0, 2π) から単位円周への（「反時計まわりの」）全単射であることを示すことができる。（連続微分可能な）曲線の長さを積分によって定義すれば、単位円周の長さが 2π であることなどがわかり、上のように定義された三角関数や円周率は、初等幾何での三角関数や円周率の素朴な定義と同じものであることが分かった。

無限乗積展開

詳細は「三角関数の無限乗積展開」を参照

三角関数は以下のように無限乗積として書ける。

${\begin{aligned}\sin \pi z&=\pi z\prod _{n=1}^{\infty }{\left(1-{\frac {z^{2}}{n^{2}}}\right)}\\\cos \pi z&=\prod _{n=1}^{\infty }\left\{1-{\frac {z^{2}}{(n-1/2)^{2}}}\right\}\end{aligned}}$

部分分数展開

詳細は「三角関数の部分分数展開」を参照

三角関数は以下のように部分分数に展開される。

${\begin{aligned}\pi \cot \pi z&=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {1}{z+n}}={\frac {1}{z}}+\sum _{n=1}^{\infty }{\frac {2z}{z^{2}-n^{2}}}\\\pi \tan \pi z&=-\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {1}{z+1/2+n}}=-\sum _{n=0}^{\infty }{\frac {2z}{z^{2}-(n+1/2)^{2}}}\\{\frac {\pi }{\sin \pi z}}&=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {(-1)^{n}}{z+n}}={\frac {1}{z}}+\sum _{n=1}^{\infty }{\frac {(-1)^{n}2z}{z^{2}-n^{2}}}\\{\frac {\pi }{\cos \pi z}}&=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {(-1)^{n}}{z+1/2+n}}=-\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n+1)}{z^{2}-(n+1/2)^{2}}}\end{aligned}}$

逆三角関数

詳細は「逆三角関数」を参照

三角関数の定義域を適当に制限したものの逆関数を逆三角関数（ぎゃくさんかくかんすう、英: inverse trigonometric function）と呼ぶ。逆三角関数は逆関数の記法に則り、元の関数の記号に −1 を右肩に付して表す。たとえば逆正弦関数（ぎゃくせいげんかんすう、英: inverse sine; インバース・サイン）は sin⁻¹x などと表す。arcsin, arccos, arctan などの記法もよく用いられる。数値計算などにおいては、これらの逆関数はさらに asin, acos, atan などと書き表される。

${\begin{aligned}x=\sin y&\iff y=\sin ^{-1}x\\x=\cos y&\iff y=\cos ^{-1}x\\x=\tan y&\iff y=\tan ^{-1}x\\x=\cot y&\iff y=\cot ^{-1}x\\x=\sec y&\iff y=\sec ^{-1}x\\x=\csc y&\iff y=\csc ^{-1}x\end{aligned}}$

である。逆関数は逆数ではないので注意したい。逆数との混乱を避けるために、逆正弦関数 sin⁻¹x を arcsin x と書く流儀もある。一般に周期関数の逆関数は多価関数になるので、通常は逆三角関数を一価連続なる枝に制限して考えることが多い。たとえば、便宜的に主値と呼ばれる枝を

${\begin{aligned}-{\frac {\pi }{2}}&\leq \sin ^{-1}x\leq {\frac {\pi }{2}}\\0&\leq \cos ^{-1}x\leq \pi \\-{\frac {\pi }{2}}&<\tan ^{-1}x<{\frac {\pi }{2}}\end{aligned}}$

のように選ぶことが多い。またこのとき、制限があることを強調するために、Sin⁻¹x, Arcsin x のように頭文字を大文字にした表記がよく用いられる。

複素関数としての三角関数

exp z, cos z, sin z の級数による定義から、オイラーの公式 exp (iz) = cos z + i sin z を導くことができる。この公式から下記の 2 つの等式

${\begin{aligned}\exp(iz)&=e^{iz}=\cos z+i\sin z,\\\exp(-iz)&=e^{-iz}=\cos z-i\sin z\end{aligned}}$

が得られるから、これを連立させて解くことにより、正弦関数・余弦関数の指数関数を用いた表現が可能となる。すなわち、

${\begin{aligned}\cos z&={\frac {e^{iz}+e^{-iz}}{2}},\\\sin z&={\frac {e^{iz}-e^{-iz}}{2i}}\end{aligned}}$

が成り立つ。この事実により、級数によらずこの等式をもって複素変数の正弦・余弦関数の定義とすることもある。また、

${\begin{aligned}\cos(iz)&={\frac {e^{-z}+e^{z}}{2}}=\cosh z,\\\sin(iz)&={\frac {e^{-z}-e^{z}}{2i}}=i\sinh z\end{aligned}}$

が成り立つ。ここで cosh z, sinh z は双曲線関数を表す。この等式は三角関数と双曲線関数の関係式と捉えることもできる。複素数 z を z = x + iy (x, y ∈ R) と表現すると、加法定理より

${\begin{aligned}\cos z&=\cos(x+iy)=\cos x\cosh y-i\sin x\sinh y,\\\sin z&=\sin(x+iy)=\sin x\cosh y+i\cos x\sinh y\end{aligned}}$

が成り立つ。

他の三角関数は csc z = 1 / sin z, sec z = 1 / cos z, tan z = sin z / cos z, cot z = cos z / sin z によって定義できる。

• cos(x + iy) の実部のグラフ

• cos(x + iy) の虚部のグラフ

• sin(x + iy) の実部のグラフ

• sin(x + iy) の虚部のグラフ

球面三角法

詳細は「球面三角法」を参照

球面の三角形 ABC の内角を a, b, c, 各頂点の対辺に関する球の中心角を α, β, γ とするとき、次のような関係が成立する。余弦公式や正弦余弦公式は式の対称性により各記号を入れ替えたものも成立する。

正弦公式

sin a : sin b : sin c = sin α : sin β : sin γ

余弦公式

cos a = −cos b cos c + sin b sin c cos α

余弦公式

cos α = cos β cos γ + sin β sin γ cos a

正弦余弦公式

sin a cos β = cos b sin c − sin b cos c cos α

参考文献

• Maor, Eli (1998). Trigonometric Delights. Princeton University Press. ISBN 978069105754-5. 
• 志賀, 浩二『数の大航海―対数の誕生と広がり』日本評論社、1999年7月。ISBN 978-4-535-78289-1。
• 高瀬, 正仁『古典的難問に学ぶ微分積分』共立出版、2013年7月。ISBN 978-4-320-11041-0。
• Vinogradov, Ivan Matveyevich (2004-09-10). The Method of Trigonometrical Sums in the Theory of Numbers (revised ed.). Dover. ISBN 978-048643878-8. 
• 黒川, 信重、小山, 信也『多重三角関数論講義』日本評論社、2010年11月8日。ISBN 978-4-535-785557。
• 杉浦, 光夫『解析入門I』東京大学出版会〈基礎数学2〉、1980年。ISBN 978-4-13-062005-5。
• 黒田, 成俊『微分積分』共立出版〈共立講座21世紀の数学 第1巻〉、2002年。ISBN 978-4320015531。
• 高木, 貞治『定本 解析概論』岩波書店、2010年、改訂第3版。ISBN 978-4000052092。
• 小平, 邦彦『解析入門I』岩波書店、2003年、軽装版。ISBN 978-4000051927。

関連項目

• 正弦定理
• 余弦定理
• 正接定理
• 球面三角法
• コサイン4乗則
• ベクトルのなす角 - cos 関数を用いて表現される。
• ドット積
• クロス積
• ベッセル関数
• sinc関数
• 指数関数
• 双曲線関数
• オイラーの公式
• 円_(数学) - 正円の三角関数との関係
• ベジエ曲線 - 三角関数のベジエ曲線による近似
• テイラー展開 - コンピュータ上での三角関数の実装に使用

外部リンク

• 三角比の近似値表
• Weisstein, Eric W. "Trigonometric Functions". MathWorld（英語）.
• 江戸時代の三角関数表(これなあに)

wiki en

[Wiki en表示]

"Cosine" redirects here. For the similarity measure, see Cosine similarity.

Trigonometry
Outline History Usage Functions (inverse) Generalized trigonometry
Reference
Identities Exact constants Tables Unit circle
Laws and theorems
Sines Cosines Tangents Cotangents Pythagorean theorem
Calculus
Trigonometric substitution Integrals (inverse functions) Derivatives

Basis of trigonometry: if two right triangles have equal acute angles, they are similar, so their side lengths are proportional. Proportionality constants are written within the image: sin θ, cos θ, tan θ, where θ is the common measure of five acute angles.

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena, through Fourier analysis.

The most widely used trigonometric functions are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used in modern mathematics.

The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. For extending these definitions to functions whose domain is the whole projectively extended real line, one can use geometrical definitions using the standard unit circle (a circle with radius 1 unit). Modern definitions express trigonometric functions as infinite series or as solutions of differential equations. This allows extending the domain of the sine and the cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane from which some isolated points are removed.

Right-angled triangle definitions

Top: Trigonometric function sin θ for selected angles θ, π − θ, π + θ, and 2π − θ in the four quadrants.
Bottom: Graph of sine function versus angle. Angles from the top panel are identified.

Plot of the six trigonometric functions and the unit circle for an angle of 0.7 radians

In this section, the same upper-case letter denotes a vertex of a triangle and the measure of the corresponding angle; the same lower case letter denotes an edge of the triangle and its length.

Given an acute angle A = θ of a right-angled triangle, the hypotenuse h is the side that connects the two acute angles. The side b adjacent to θ is the side of the triangle that connects θ to the right angle. The third side θ is said opposite to θ.

If the angle θ is given, then all sides of the right-angled triangle are well defined up to a scaling factor. This means that the ratio of any two side lengths depends only on θ. These six ratios define thus six functions of θ, which are the trigonometric functions. More precisely, the six trigonometric functions are:

sine

$\sin \theta ={\frac {a}{h}}={\frac {\mathrm {opposite} }{\mathrm {hypotenuse} }}$

cosine

$\cos \theta ={\frac {b}{h}}={\frac {\mathrm {adjacent} }{\mathrm {hypotenuse} }}$

tangent

$\tan \theta ={\frac {a}{b}}={\frac {\mathrm {opposite} }{\mathrm {adjacent} }}$

cosecant

$\csc \theta ={\frac {h}{a}}={\frac {\mathrm {hypotenuse} }{\mathrm {opposite} }}$

secant

$\sec \theta ={\frac {h}{b}}={\frac {\mathrm {hypotenuse} }{\mathrm {adjacent} }}$

cotangent

$\cot \theta ={\frac {b}{a}}={\frac {\mathrm {adjacent} }{\mathrm {opposite} }}$

In a right angled triangle, the sum of the two acute angles is a right angle, that is 90° or ${\textstyle {\frac {\pi }{2}}}$ radians.

Summary of relationships between trigonometric functions

Function	Abbreviation	Description	Relationship
			using radians	using degrees
sine	sin	opposite/hypotenuse	$$ sin ⁡ θ = cos ⁡ ( π 2 − θ ) = 1 csc ⁡ θ {\displaystyle \sin \theta =\cos \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\csc \theta }}}	$$ sin ⁡ x = cos ⁡ ( 90 ∘ − x ) = 1 csc ⁡ x {\displaystyle \sin x=\cos \left(90^{\circ }-x\right)={\frac {1}{\csc x}}}
cosine	cos	adjacent/hypotenuse	$$ cos ⁡ θ = sin ⁡ ( π 2 − θ ) = 1 sec ⁡ θ {\displaystyle \cos \theta =\sin \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\sec \theta }}\,}	$$ cos ⁡ x = sin ⁡ ( 90 ∘ − x ) = 1 sec ⁡ x {\displaystyle \cos x=\sin \left(90^{\circ }-x\right)={\frac {1}{\sec x}}\,}
tangent	tan (or tg)	opposite/adjacent	$$ tan ⁡ θ = sin ⁡ θ cos ⁡ θ = cot ⁡ ( π 2 − θ ) = 1 cot ⁡ θ {\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}=\cot \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cot \theta }}}	$$ tan ⁡ x = sin ⁡ x cos ⁡ x = cot ⁡ ( 90 ∘ − x ) = 1 cot ⁡ x {\displaystyle \tan x={\frac {\sin x}{\cos x}}=\cot \left(90^{\circ }-x\right)={\frac {1}{\cot x}}}
cotangent	cot (or cotan or cotg or ctg or ctn)	adjacent/opposite	$$ cot ⁡ θ = cos ⁡ θ sin ⁡ θ = tan ⁡ ( π 2 − θ ) = 1 tan ⁡ θ {\displaystyle \cot \theta ={\frac {\cos \theta }{\sin \theta }}=\tan \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\tan \theta }}}	$$ cot ⁡ x = cos ⁡ x sin ⁡ x = tan ⁡ ( 90 ∘ − x ) = 1 tan ⁡ x {\displaystyle \cot x={\frac {\cos x}{\sin x}}=\tan \left(90^{\circ }-x\right)={\frac {1}{\tan x}}}
secant	sec	hypotenuse/adjacent	$$ sec ⁡ θ = csc ⁡ ( π 2 − θ ) = 1 cos ⁡ θ {\displaystyle \sec \theta =\csc \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cos \theta }}}	$$ sec ⁡ x = csc ⁡ ( 90 ∘ − x ) = 1 cos ⁡ x {\displaystyle \sec x=\csc \left(90^{\circ }-x\right)={\frac {1}{\cos x}}}
cosecant	csc (or cosec)	hypotenuse/opposite	$$ csc ⁡ θ = sec ⁡ ( π 2 − θ ) = 1 sin ⁡ θ {\displaystyle \csc \theta =\sec \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\sin \theta }}}	$$ csc ⁡ x = sec ⁡ ( 90 ∘ − x ) = 1 sin ⁡ x {\displaystyle \csc x=\sec \left(90^{\circ }-x\right)={\frac {1}{\sin x}}}

Radians versus degrees

In geometric applications, the argument of a trigonometric function is generally the measure of an angle. For this purpose, any angular unit is convenient, and angles are most commonly measured in degrees.

When using trigonometric function in calculus, their argument is generally not an angle, but rather a real number. In this case, it is more suitable to express the argument of the trigonometric as the length of the arc of the unit circle delimited by an angle with the center of the circle as vertex. Therefore, one uses the radian as angular unit: a radian is the angle that delimits an arc of length 1 on the unit circle. A complete turn is thus an angle of 2π radians.

A great advantage of radians is that many formulas are much simpler when using them, typically all formulas relative to derivatives and integrals.

This is thus a general convention that, when the angular unit is not explicitly specified, the arguments of trigonometric functions are always expressed in radians.

Unit-circle definitions

In this illustration, the six trigonometric functions of an arbitrary angle θ are represented as Cartesian coordinates of points related to the unit circle. The ordinates of A, B and D are sin θ, tan θ and csc θ, respectively, while the abscissas of A, C and E are cos θ, cot θ and sec θ, respectively.

Signs of trigonometric functions in each quadrant. The mnemonic "all science teachers (are) crazy" lists the functions which are positive from quadrants I to IV. This is a variation on the mnemonic "All Students Take Calculus".

The six trigonometric functions can be defined as coordinate values of points on the Euclidean plane that are related to the unit circle, which is the circle of radius one centered at the origin O of this coordinate system. While right-angled triangle definitions permit the definition of the trigonometric functions for angles between 0 and ${\textstyle {\frac {\pi }{2}}}$ radian (90°), the unit circle definitions allow to extend the domain of the trigonometric functions to all positive and negative real numbers.

Rotating a ray from the direction of the positive half of the x-axis by an angle θ (counterclockwise for $$

 
   
     
       θ
       >
       0
       ,
     
   
   {\displaystyle \theta >0,}
 

and clockwise for

 
   
     
       θ
       <
       0
     
   
   {\displaystyle \theta <0}
 

) yields intersection points of this ray (see the figure) with the unit circle: $$

 
   
     
       
         A
       
       =
       (
       
         x
         
           
             A
           
         
       
       ,
       
         y
         
           
             A
           
         
       
       )
     
   
   {\displaystyle \mathrm {A} =(x_{\mathrm {A} },y_{\mathrm {A} })}
 

, and, by extending the ray to a line if necessary, with the line $$

 
   
     
       
         “
       
       x
       =
       1
       
         ”
       
       :
       
       
         B
       
       =
       (
       
         x
         
           
             B
           
         
       
       ,
       
         y
         
           
             B
           
         
       
       )
       ,
     
   
   {\displaystyle {\text{“}}x=1{\text{”}}:\;\mathrm {B} =(x_{\mathrm {B} },y_{\mathrm {B} }),}
 

and with the line $$

 
   
     
       
         “
       
       y
       =
       1
       
         ”
       
       :
       
       
         C
       
       =
       (
       
         x
         
           
             C
           
         
       
       ,
       
         y
         
           
             C
           
         
       
       )
       .
     
   
   {\displaystyle {\text{“}}y=1{\text{”}}:\;\mathrm {C} =(x_{\mathrm {C} },y_{\mathrm {C} }).}
 

The tangent line to the unit circle in point A, which is orthogonal to this ray, intersects the y- and x-axis in points $$

 
   
     
       
         D
       
       =
       (
       0
       ,
       
         y
         
           
             D
           
         
       
       )
     
   
   {\displaystyle \mathrm {D} =(0,y_{\mathrm {D} })}
 

and $$

 
   
     
       
         E
       
       =
       (
       
         x
         
           
             E
           
         
       
       ,
       0
       )
     
   
   {\displaystyle \mathrm {E} =(x_{\mathrm {E} },0)}
 

. The coordinate values of these points give all the existing values of the trigonometric functions for arbitrary real values of θ in the following manner.

The trigonometric functions cos and sin are defined, respectively, as the x- and y-coordinate values of point A, i.e.,

$\cos \theta =x_{\mathrm {A} }\quad$ and $\quad \sin \theta =y_{\mathrm {A} }.$

In the range $0\leq \theta \leq \pi /2$ this definition coincides with the right-angled triangle definition by taking the right-angled triangle to have the unit radius OA as hypotenuse, and since for all points $\mathrm {P} =(x,y)$ on the unit circle the equation $x^{2}+y^{2}=1$ holds, this definition of cosine and sine also satisfies the Pythagorean identity

$\cos ^{2}\theta +\sin ^{2}\theta =1.$

The other trigonometric functions can be found along the unit circle as

$\tan \theta =y_{\mathrm {B} }\quad$ and $\quad \cot \theta =x_{\mathrm {C} },$

$\csc \theta \ =y_{\mathrm {D} }\quad$ and $\quad \sec \theta =x_{\mathrm {E} }.$

By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is

$\tan \theta ={\frac {\sin \theta }{\cos \theta }},\quad \cot \theta ={\frac {\cos \theta }{\sin \theta }},\quad \sec \theta ={\frac {1}{\cos \theta }},\quad \csc \theta ={\frac {1}{\sin \theta }}.$

Trigonometric functions: Sine, Cosine, Tangent, Cosecant (dotted), Secant (dotted), Cotangent (dotted)

As a rotation of an angle of $\pm 2\pi$ does not change the position or size of a shape, the points A, B, C, D, and E are the same for two angles whose difference is an integer multiple of $2\pi$. Thus trigonometric functions are periodic functions with period $2\pi$. That is, the equalities

$\sin \theta =\sin \left(\theta +2k\pi \right)\quad$ and $\quad \cos \theta =\cos \left(\theta +2k\pi \right)$

hold for any angle θ and any integer k. The same is true for the four other trigonometric functions. Observing the sign and the monotonicity of the functions sine, cosine, cosecant, and secant in the four quadrants, shows that 2π is the smallest value for which they are periodic, i.e., 2π is the fundamental period of these functions. However, already after a rotation by an angle $\pi$ the points B and C return to their original position, so that the tangent function and the cotangent function have a fundamental period of π. That is, the equalities

$\tan \theta =\tan(\theta +k\pi )\quad$ and $\quad \cot \theta =\cot(\theta +k\pi )$

hold for any angle θ and any integer k.

Algebraic values

The unit circle, with some points labeled with their cosine and sine (in this order), and the corresponding angles in radians and degrees.

The algebraic expressions for the most important angles are as follows:

$\sin 0=\sin 0^{\circ }\quad ={\frac {\sqrt {0}}{2}}=0$ (straight angle)

$\sin {\frac {\pi }{6}}=\sin 30^{\circ }={\frac {\sqrt {1}}{2}}={\frac {1}{2}}$

$\sin {\frac {\pi }{4}}=\sin 45^{\circ }={\frac {\sqrt {2}}{2}}={\frac {\sqrt {2}}{2}}$

$\sin {\frac {\pi }{3}}=\sin 60^{\circ }={\frac {\sqrt {3}}{2}}={\frac {\sqrt {3}}{2}}$

$\sin {\frac {\pi }{2}}=\sin 90^{\circ }={\frac {\sqrt {4}}{2}}=1$ (right angle)

Writing the numerators as square roots of consecutive non-negative integers, with a denominator of 2, provides an easy way to remember the values.

Such simple expressions generally do not exist for other angles which are rational multiples of a straight angle.

For an angle which, measured in degrees, is a multiple of three, the sine and the cosine may be expressed in terms of square roots, see Trigonometric constants expressed in real radicals. These values of the sine and the cosine may thus be constructed by ruler and compass.

For an angle of an integer number of degrees, the sine and the cosine may be expressed in terms of square roots and the cube root of a non-real complex number. Galois theory allows proving that, if the angle is not a multiple of 3°, non-real cube roots are unavoidable.

For an angle which, measured in degrees, is a rational number, the sine and the cosine are algebraic numbers, which may be expressed in terms of nth roots. This results from the fact that the Galois groups of the cyclotomic polynomials are cyclic.

For an angle which, measured in degrees, is not a rational number, then either the angle or both the sine and the cosine are transcendental numbers. This is a corollary of Baker's theorem, proved in 1966.

Simple algebraic values

The following table summarizes the simplest algebraic values of trigonometric functions. The symbol ∞ represents the point at infinity on the projectively extended real line; it is not signed, because, when it appears in the table, the corresponding trigonometric function tends to +∞ on one side, and to –∞ on the other side, when the argument tends to the value in the table.

${\begin{array}{|c|ccccccccc|}\hline {\begin{matrix}{\text{Radian}}\\{\text{Degree}}\end{matrix}}&{\begin{matrix}0\\0^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{12}}\\15^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{8}}\\22.5^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{6}}\\30^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{4}}\\45^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{3}}\\60^{\circ }\end{matrix}}&{\begin{matrix}{\frac {3\pi }{8}}\\67.5^{\circ }\end{matrix}}&{\begin{matrix}{\frac {5\pi }{12}}\\75^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{2}}\\90^{\circ }\end{matrix}}\\\hline \sin &0&{\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}&{\frac {\sqrt {2-{\sqrt {2}}}}{2}}&{\frac {1}{2}}&{\frac {\sqrt {2}}{2}}&{\frac {\sqrt {3}}{2}}&{\frac {\sqrt {2+{\sqrt {2}}}}{2}}&{\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}&1\\\cos &1&{\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}&{\frac {\sqrt {2+{\sqrt {2}}}}{2}}&{\frac {\sqrt {3}}{2}}&{\frac {\sqrt {2}}{2}}&{\frac {1}{2}}&{\frac {\sqrt {2-{\sqrt {2}}}}{2}}&{\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}&0\\\tan &0&2-{\sqrt {3}}&{\sqrt {2}}-1&{\frac {\sqrt {3}}{3}}&1&{\sqrt {3}}&{\sqrt {2}}+1&2+{\sqrt {3}}&\infty \\\cot &\infty &2+{\sqrt {3}}&{\sqrt {2}}+1&{\sqrt {3}}&1&{\frac {\sqrt {3}}{3}}&{\sqrt {2}}-1&2-{\sqrt {3}}&0\\\sec &1&{\sqrt {6}}-{\sqrt {2}}&{\sqrt {2}}{\sqrt {2-{\sqrt {2}}}}&{\frac {2{\sqrt {3}}}{3}}&{\sqrt {2}}&2&{\sqrt {2}}{\sqrt {2+{\sqrt {2}}}}&{\sqrt {6}}+{\sqrt {2}}&\infty \\\csc &\infty &{\sqrt {6}}+{\sqrt {2}}&{\sqrt {2}}{\sqrt {2+{\sqrt {2}}}}&2&{\sqrt {2}}&{\frac {2{\sqrt {3}}}{3}}&{\sqrt {2}}{\sqrt {2-{\sqrt {2}}}}&{\sqrt {6}}-{\sqrt {2}}&1\\\hline \end{array}}$

In calculus

The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full cycle centered on the origin.

Animation for the approximation of cosine via Taylor polynomials.

$$

 
   
     
       cos
       ⁡
       (
       x
       )
     
   
   {\displaystyle \cos(x)}
 

together with the first Taylor polynomials $$

 
   
     
       
         p
         
           n
         
       
       (
       x
       )
       =
       
         ∑
         
           k
           =
           0
         
         
           n
         
       
       (
       −
       1
       
         )
         
           k
         
       
       
         
           
             x
             
               2
               k
             
           
           
             (
             2
             k
             )
             !
           
         
       
     
   
   {\displaystyle p_{n}(x)=\sum _{k=0}^{n}(-1)^{k}{\frac {x^{2k}}{(2k)!}}}
 

Trigonometric functions are differentiable. This is not immediately evident from the above geometrical definitions. Moreover, the modern trend in mathematics is to build geometry from calculus rather than the converse. Therefore, except at a very elementary level, trigonometric functions are defined using the methods of calculus.

For defining trigonometric functions inside calculus, there are two equivalent possibilities, either using power series or differential equations. These definitions are equivalent, as starting from one of them, it is easy to retrieve the other as a property. However the definition through differential equations is somehow more natural, since, for example, the choice of the coefficients of the power series may appear as quite arbitrary, and the Pythagorean identity is much easier to deduce from the differential equations.

Definition by differential equations

Sine and cosine are the unique differentiable functions such that

${\begin{aligned}{\frac {d}{dx}}\sin x&=\cos x,\\{\frac {d}{dx}}\cos x&=-\sin x,\\\sin 0&=0,\\\cos 0&=1.\end{aligned}}$

Differentiating these equations, one gets that both sine and cosine are solutions of the differential equation

$y''+y=0.$

Applying the quotient rule to the definition of the tangent as the quotient of the sine by the cosine, one gets that the tangent function verifies

${\frac {d}{dx}}\tan x=1+\tan ^{2}x.$

Power series expansion

Applying the differential equations to power series with indeterminate coefficients, one may deduce recurrence relations for the coefficients of the Taylor series of the sine and cosine functions. These recurrence relations are easy to solve, and give the series expansions

${\begin{aligned}\sin x&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \\[8pt]&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}\\[8pt]\cos x&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \\[8pt]&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}.\end{aligned}}$

The radius of convergence of these series is infinite. Therefore, the sine and the cosine can be extended to entire functions (also called "sine" and "cosine"), which are (by definition) complex-valued functions that are defined and holomorphic on the whole complex plane.

Being defined as fractions of entire functions, the other trigonometric functions may be extended to meromorphic functions, that is functions that are holomorphic in the whole complex plane, except some isolated points called poles. Here, the poles are the numbers of the form $$

 
   
     
       (
       2
       k
       +
       1
       )
       
         
           π
           2
         
       
     
   
   {\textstyle (2k+1){\frac {\pi }{2}}}
 

for the tangent and the secant, or $$

 
   
     
       k
       π
     
   
   {\displaystyle k\pi }
 

for the cotangent and the cosecant, where k is an arbitrary integer.

Recurrences relations may also be computed for the coefficients of the Taylor series of the other trigonometric functions. These series have a finite radius of convergence. Their coefficients have a combinatorial interpretation: they enumerate alternating permutations of finite sets.

More precisely, defining

U_n, the nth up/down number,

B_n, the nth Bernoulli number, and

E_n, is the nth Euler number,

one has the following series expansions:

${\begin{aligned}\tan x&{}=\sum _{n=0}^{\infty }{\frac {U_{2n+1}x^{2n+1}}{(2n+1)!}}\\&{}=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}2^{2n}\left(2^{2n}-1\right)B_{2n}x^{2n-1}}{(2n)!}}\\&{}=x+{\frac {1}{3}}x^{3}+{\frac {2}{15}}x^{5}+{\frac {17}{315}}x^{7}+\cdots ,\qquad {\text{for }}|x|<{\frac {\pi }{2}}.\end{aligned}}$

${\begin{aligned}\csc x&{}=\sum _{n=0}^{\infty }{\frac {(-1)^{n+1}2\left(2^{2n-1}-1\right)B_{2n}x^{2n-1}}{(2n)!}}\\&{}=x^{-1}+{\frac {1}{6}}x+{\frac {7}{360}}x^{3}+{\frac {31}{15120}}x^{5}+\cdots ,\qquad {\text{for }}0<|x|<\pi .\end{aligned}}$

${\begin{aligned}\sec x&{}=\sum _{n=0}^{\infty }{\frac {U_{2n}x^{2n}}{(2n)!}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}E_{2n}x^{2n}}{(2n)!}}\\&{}=1+{\frac {1}{2}}x^{2}+{\frac {5}{24}}x^{4}+{\frac {61}{720}}x^{6}+\cdots ,\qquad {\text{for }}|x|<{\frac {\pi }{2}}.\end{aligned}}$

${\begin{aligned}\cot x&{}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}2^{2n}B_{2n}x^{2n-1}}{(2n)!}}\\&{}=x^{-1}-{\frac {1}{3}}x-{\frac {1}{45}}x^{3}-{\frac {2}{945}}x^{5}-\cdots ,\qquad {\text{for }}0<|x|<\pi .\end{aligned}}$

There is a series representation as partial fraction expansion where just translated reciprocal functions are summed up, such that the poles of the cotangent function and the reciprocal functions match:

$\pi \cot \pi x=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {1}{x+n}}.$

This identity can be proven with the Herglotz trick. Combining the (–n)th with the nth term lead to absolutely convergent series:

$\pi \cot \pi x={\frac {1}{x}}+2x\sum _{n=1}^{\infty }{\frac {1}{x^{2}-n^{2}}}\ ,\quad {\frac {\pi }{\sin \pi x}}={\frac {1}{x}}+2x\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{x^{2}-n^{2}}}.$

Infinite product expansion

The following infinite product for the sine is of great importance in complex anaylsis:

$\sin z=z\displaystyle \prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{n^{2}\pi ^{2}}}\right),\quad z\in \mathbb {C} .$

For the proof of this expansion, see Sine. From this, it can be deduced that

$\cos z=\displaystyle \prod _{n=1}^{\infty }\left(1-{\frac {z^{2}}{\left(n-{\frac {1}{2}}\right)^{2}\pi ^{2}}}\right),\quad z\in \mathbb {C} .$

Relationship to exponential function (Euler's formula)

$$

 
   
     
       cos
       ⁡
       (
       θ
       )
     
   
   {\displaystyle \cos(\theta )}
 

and $$

 
   
     
       sin
       ⁡
       (
       θ
       )
     
   
   {\displaystyle \sin(\theta )}
 

are the real and imaginary part of $$

 
   
     
       
         e
         
           i
           θ
         
       
     
   
   {\displaystyle e^{i\theta }}
 

respectively.

Euler's formula relates sine and cosine to the exponential function:

$e^{ix}=\cos x+i\sin x.$

This formula is commonly considered for real values of x, but it remains true for all complex values.

Proof: Let $$

 
   
     
       
         f
         
           1
         
       
       (
       x
       )
       =
       cos
       ⁡
       x
       +
       i
       sin
       ⁡
       x
       ,
     
   
   {\displaystyle f_{1}(x)=\cos x+i\sin x,}
 

and $$

 
   
     
       
         f
         
           2
         
       
       (
       x
       )
       =
       
         e
         
           i
           x
         
       
       .
     
   
   {\displaystyle f_{2}(x)=e^{ix}.}
 

One has $$

 
   
     
       
         
           d
           
             d
             x
           
         
       
       
         f
         
           j
         
       
       (
       x
       )
       =
       i
       
         f
         
           j
         
       
       (
       x
       )
     
   
   {\textstyle {\frac {d}{dx}}f_{j}(x)=if_{j}(x)}
 

for j = 1, 2. The quotient rule implies thus that $$

 
   
     
       
         
           d
           
             d
             x
           
         
       
       
         (
         
           
             
               
                 f
                 
                   1
                 
               
               (
               x
               )
             
             
               
                 f
                 
                   2
                 
               
               (
               x
               )
             
           
         
         )
       
       =
       0
     
   
   {\textstyle {\frac {d}{dx}}\left({\frac {f_{1}(x)}{f_{2}(x)}}\right)=0}
 

. Therefore, $$

 
   
     
       
         
           
             
               f
               
                 1
               
             
             (
             x
             )
           
           
             
               f
               
                 2
               
             
             (
             x
             )
           
         
       
     
   
   {\textstyle {\frac {f_{1}(x)}{f_{2}(x)}}}
 

is a constant function, which equals 1, as $$

 
   
     
       
         f
         
           1
         
       
       (
       0
       )
       =
       
         f
         
           2
         
       
       (
       0
       )
       =
       1.
     
   
   {\displaystyle f_{1}(0)=f_{2}(0)=1.}
 

This proves the formula.

One has

${\begin{aligned}e^{ix}&=\cos x+i\sin x\\[5pt]e^{-ix}&=\cos x-i\sin x.\end{aligned}}$

Solving this linear system in sine and cosine, one can express them in terms of the exponential function:

${\begin{aligned}\sin x&={\frac {e^{ix}-e^{-ix}}{2i}}\\[5pt]\cos x&={\frac {e^{ix}+e^{-ix}}{2}}.\end{aligned}}$

When x is real, this may be rewritten as

$\cos x=\operatorname {Re} \left(e^{ix}\right),\qquad \sin x=\operatorname {Im} \left(e^{ix}\right).$

Most trigonometric identities can be proved by expressing trigonometric functions in terms of the complex exponential function by using above formulas, and then using the identity $e^{a+b}=e^{a}e^{b}$ for simplifying the result.

Definitions using functional equations

One can also define the trigonometric functions using various functional equations.

For example, the sine and the cosine form the unique pair of continuous functions that satisfy the difference formula

$\cos(x-y)=\cos x\cos y+\sin x\sin y\,$

and the added condition

$0<x\cos x<\sin x<x\quad {\text{ for }}\quad 0<x<1.$

In the complex plane

The sine and cosine of a complex number $z=x+iy$ can be expressed in terms of real sines, cosines, and hyperbolic functions as follows:

${\begin{aligned}\sin z&=\sin x\cosh y+i\cos x\sinh y\\[5pt]\cos z&=\cos x\cosh y-i\sin x\sinh y\end{aligned}}$

By taking advantage of domain coloring, it is possible to graph the trigonometric functions as complex-valued functions. Various features unique to the complex functions can be seen from the graph; for example, the sine and cosine functions can be seen to be unbounded as the imaginary part of $z$ becomes larger (since the color white represents infinity), and the fact that the functions contain simple zeros or poles is apparent from the fact that the hue cycles around each zero or pole exactly once. Comparing these graphs with those of the corresponding Hyperbolic functions highlights the relationships between the two.

Trigonometric functions in the complex plane

$$ sin ⁡ z {\displaystyle \sin z\,}	$$ cos ⁡ z {\displaystyle \cos z\,}	$$ tan ⁡ z {\displaystyle \tan z\,}	$$ cot ⁡ z {\displaystyle \cot z\,}	$$ sec ⁡ z {\displaystyle \sec z\,}	$$ csc ⁡ z {\displaystyle \csc z\,}

Basic identities

Many identities interrelate the trigonometric functions. This section contains the most basic ones; for more identities, see List of trigonometric identities. These identities may be proved geometrically from the unit-circle definitions or the right-angled-triangle definitions (although, for the latter definitions, care must be taken for angles that are not in the interval [0, π/2], see Proofs of trigonometric identities). For non-geometrical proofs using only tools of calculus, one may use directly the differential equations, in a way that is similar to that of the above proof of Euler's identity. One can also use Euler's identity for expressing all trigonometric functions in terms of complex exponentials and using properties of the exponential function.

Parity

The cosine and the secant are even functions; the other trigonometric functions are odd functions. That is:

${\begin{aligned}\sin(-x)&=-\sin x\\\cos(-x)&=\cos x\\\tan(-x)&=-\tan x\\\cot(-x)&=-\cot x\\\csc(-x)&=-\csc x\\\sec(-x)&=\sec x.\end{aligned}}$

Periods

All trigonometric functions are periodic functions of period 2π. This is the smallest period, except for the tangent and the cotangent, which have π as smallest period. This means that, for every integer k, one has

${\begin{aligned}\sin(x+2k\pi )&=\sin x\\\cos(x+2k\pi )&=\cos x\\\tan(x+k\pi )&=\tan x\\\cot(x+k\pi )&=\cot x\\\csc(x+2k\pi )&=\csc x\\\sec(x+2k\pi )&=\sec x.\end{aligned}}$

Pythagorean identity

The Pythagorean identity, is the expression of the Pythagorean theorem in terms of trigonometric functions. It is

$\sin ^{2}x+\cos ^{2}x=1.$

Sum and difference formulas

The sum and difference formulas allow expanding the sine, the cosine, and the tangent of a sum or a difference of two angles in terms of sines and cosines and tangents of the angles themselves. These can be derived geometrically, using arguments that date to Ptolemy. One can also produce them algebraically using Euler's formula.

Sum

${\begin{aligned}\sin \left(x+y\right)&=\sin x\cos y+\cos x\sin y,\\\cos \left(x+y\right)&=\cos x\cos y-\sin x\sin y,\\\tan(x+y)&={\frac {\tan x+\tan y}{1-\tan x\tan y}}.\end{aligned}}$

Difference

${\begin{aligned}\sin \left(x-y\right)&=\sin x\cos y-\cos x\sin y,\\\cos \left(x-y\right)&=\cos x\cos y+\sin x\sin y,\\\tan(x-y)&={\frac {\tan x-\tan y}{1+\tan x\tan y}}.\end{aligned}}$

When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulae.

${\begin{aligned}\sin 2x&=2\sin x\cos x={\frac {2\tan x}{1+\tan ^{2}x}},\\\cos 2x&=\cos ^{2}x-\sin ^{2}x=2\cos ^{2}x-1=1-2\sin ^{2}x={\frac {1-\tan ^{2}x}{1+\tan ^{2}x}},\\\tan 2x&={\frac {2\tan x}{1-\tan ^{2}x}}.\end{aligned}}$

These identities can be used to derive the product-to-sum identities.

By setting $$

 
   
     
       θ
       =
       2
       x
     
   
   {\displaystyle \theta =2x}
 

and $$

 
   
     
       t
       =
       tan
       ⁡
       x
       ,
     
   
   {\displaystyle t=\tan x,}
 

this allows expressing all trigonometric functions of $$

 
   
     
       θ
     
   
   {\displaystyle \theta }
 

as a rational fraction of $$

 
   
     
       t
       =
       tan
       ⁡
       
         
           θ
           2
         
       
     
   
   {\textstyle t=\tan {\frac {\theta }{2}}}
 

:

${\begin{aligned}\sin \theta &={\frac {2t}{1+t^{2}}},\\\cos \theta &={\frac {1-t^{2}}{1+t^{2}}},\\\tan \theta &={\frac {2t}{1-t^{2}}}.\end{aligned}}$

Together with

$d\theta ={\frac {2}{1+t^{2}}}\,dt,$

this is the tangent half-angle substitution, which allows reducing the computation of integrals and antiderivatives of trigonometric functions to that of rational fractions.

Derivatives and antiderivatives

The derivatives of trigonometric functions result from those of sine and cosine by applying quotient rule. The values given for the antiderivatives in the following table can be verified by differentiating them. The number C is a constant of integration.

${\begin{array}{|c|c|c|}\hline f(x)&f'(x)&\int f(x)\,dx\\\hline \sin x&\cos x&-\cos x+C\\\cos x&-\sin x&\sin x+C\\\tan x&\sec ^{2}x=1+\tan ^{2}x&-\ln \left(|\cos x|\right)+C\\\cot x&-\csc ^{2}x=-(1+\cot ^{2}x)&\ln \left(|\sin x|\right)+C\\\sec x&\sec x\tan x&\ln \left(|\sec x+\tan x|\right)+C\\\csc x&-\csc x\cot x&-\ln \left(|\csc x+\cot x|\right)+C\\\hline \end{array}}$

Inverse functions

Main article: Inverse trigonometric functions

The trigonometric functions are periodic, and hence not injective, so strictly speaking, they do not have an inverse function. However, on each interval on which a trigonometric function is monotonic, one can define an inverse function, and this defines inverse trigonometric functions as multivalued functions. To define a true inverse function, one must restrict the domain to an interval where the function is monotonic, and is thus bijective from this interval to its image by the function. The common choice for this interval, called the set of principal values, is given in the following table. As usual, the inverse trigonometric functions are denoted with the prefix "arc" before the name or its abbreviation of the function.

${\begin{array}{|c|c|c|c|}\hline {\text{Function}}&{\text{Definition}}&{\text{Domain}}&{\text{Set of principal values}}\\\hline y=\arcsin x&\sin y=x&-1\leq x\leq 1&-{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}}\\y=\arccos x&\cos y=x&-1\leq x\leq 1&0\leq y\leq \pi \\y=\arctan x&\tan y=x&-\infty \leq x\leq \infty &-{\frac {\pi }{2}}<y<{\frac {\pi }{2}}\\y=\operatorname {arccot} x&\cot y=x&-\infty \leq x\leq \infty &0<y<\pi \\y=\operatorname {arcsec} x&\sec y=x&x<-1{\text{ or }}x>1&0\leq y\leq \pi ,\;y\neq {\frac {\pi }{2}}\\y=\operatorname {arccsc} x&\csc y=x&x<-1{\text{ or }}x>1&-{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}},\;y\neq 0\\\hline \end{array}}$

The notations sin^−1, cos^−1, etc. are often used for arcsin and arccos, etc. When this notation is used, inverse functions could be confused with multiplicative inverses. The notation with the "arc" prefix avoids such a confusion, though "arcsec" for arcsecant can be confused with "arcsecond".

Just like the sine and cosine, the inverse trigonometric functions can also be expressed in terms of infinite series. They can also be expressed in terms of complex logarithms. See Inverse trigonometric functions for details.

Applications

Main article: Uses of trigonometry

Angles and sides of a triangle

In this sections A, B, C denote the three (interior) angles of a triangle, and a, b, c denote the lengths of the respective opposite edges. They are related by various formulas, which are named by the trigonometric functions they involve.

Law of sines

The law of sines states that for an arbitrary triangle with sides a, b, and c and angles opposite those sides A, B and C:

${\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}={\frac {2\Delta }{abc}},$

where Δ is the area of the triangle, or, equivalently,

${\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R,$

where R is the triangle's circumradius.

It can be proven by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in triangulation, a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.

Law of cosines

The law of cosines (also known as the cosine formula or cosine rule) is an extension of the Pythagorean theorem:

$c^{2}=a^{2}+b^{2}-2ab\cos C,\,$

or equivalently,

$\cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}.$

In this formula the angle at C is opposite to the side c. This theorem can be proven by dividing the triangle into two right ones and using the Pythagorean theorem.

The law of cosines can be used to determine a side of a triangle if two sides and the angle between them are known. It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all the sides are known.

Law of tangents

Main article: Law of tangents

The following all form the law of tangents

${\frac {\tan {\frac {A-B}{2}}}{\tan {\frac {A+B}{2}}}}={\frac {a-b}{a+b}}\,;\qquad {\frac {\tan {\frac {A-C}{2}}}{\tan {\frac {A+C}{2}}}}={\frac {a-c}{a+c}}\,;\qquad {\frac {\tan {\frac {B-C}{2}}}{\tan {\frac {B+C}{2}}}}={\frac {b-c}{b+c}}.$

The explanation of the formulae in words would be cumbersome, but the patterns of sums and differences, for the lengths and corresponding opposite angles, are apparent in the theorem.

Law of cotangents

Main article: Law of cotangents

If

$\zeta ={\sqrt {{\frac {1}{s}}(s-a)(s-b)(s-c)}}~$ (the radius of the inscribed circle for the triangle)

and

$s={\frac {a+b+c}{2}}~$ (the semi-perimeter for the triangle),

then the following all form the law of cotangents

$\cot {\frac {A}{2}}={\frac {s-a}{\zeta }}~;\qquad \cot {\frac {B}{2}}={\frac {s-b}{\zeta }}~;\qquad \cot {\frac {C}{2}}={\frac {s-c}{\zeta }}~.$

It follows that

${\frac {\cot {\dfrac {A}{2}}}{s-a}}={\frac {\cot {\dfrac {B}{2}}}{s-b}}={\frac {\cot {\dfrac {C}{2}}}{s-c}}={\frac {1}{\zeta }}~.$

In words the theorem is: the cotangent of a half-angle equals the ratio of the semi-perimeter minus the opposite side to the said angle, to the inradius for the triangle.

A Lissajous curve, a figure formed with a trigonometry-based function.

Periodic functions

An animation of the additive synthesis of a square wave with an increasing number of harmonics

Sinusoidal basis functions (bottom) can form a sawtooth wave (top) when added. All the basis functions have nodes at the nodes of the sawtooth, and all but the fundamental (k = 1) have additional nodes. The oscillation seen about the sawtooth when k is large is called the Gibbs phenomenon

The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describe simple harmonic motion, which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. The sine and cosine functions are one-dimensional projections of uniform circular motion.

Trigonometric functions also prove to be useful in the study of general periodic functions. The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light waves.

Under rather general conditions, a periodic function f(x) can be expressed as a sum of sine waves or cosine waves in a Fourier series. Denoting the sine or cosine basis functions by φ_k, the expansion of the periodic function f(t) takes the form:

$f(t)=\sum _{k=1}^{\infty }c_{k}\varphi _{k}(t).$

For example, the square wave can be written as the Fourier series

$f_{\text{square}}(t)={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\sin {\big (}(2k-1)t{\big )} \over 2k-1}.$

In the animation of a square wave at top right it can be seen that just a few terms already produce a fairly good approximation. The superposition of several terms in the expansion of a sawtooth wave are shown underneath.

History

Main article: History of trigonometric functions

While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period. The chord function was discovered by Hipparchus of Nicaea (180–125 BCE) and Ptolemy of Roman Egypt (90–165 CE). The functions of sine and versine (1 - cosine) can be traced back to the jyā and koti-jyā functions used in Gupta period Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin. (See Aryabhata's sine table.)

All six trigonometric functions in current use were known in Islamic mathematics by the 9th century, as was the law of sines, used in solving triangles. With the exception of the sine (which was adopted from Indian mathematics), the other five modern trigonometric functions were discovered by Arabic mathematicians, including the cosine, tangent, cotangent, secant and cosecant. Al-Khwārizmī (c. 780–850) produced tables of sines, cosines and tangents. Circa 830, Habash al-Hasib al-Marwazi discovered the cotangent, and produced tables of tangents and cotangents. Muhammad ibn Jābir al-Harrānī al-Battānī (853–929) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°. The trigonometric functions were later studied by mathematicians including Omar Khayyám, Bhāskara II, Nasir al-Din al-Tusi, Jamshīd al-Kāshī (14th century), Ulugh Beg (14th century), Regiomontanus (1464), Rheticus, and Rheticus' student Valentinus Otho.

Madhava of Sangamagrama (c. 1400) made early strides in the analysis of trigonometric functions in terms of infinite series. (See Madhava series and Madhava's sine table.)

The terms tangent and secant were first introduced by the Danish mathematician Thomas Fincke in his book Geometria rotundi (1583).

The 16th century French mathematician Albert Girard made the first published use of the abbreviations sin, cos, and tan in his book Trigonométrie.

In a paper published in 1682, Leibniz proved that sin x is not an algebraic function of x.

Leonhard Euler's Introductio in analysin infinitorum (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, also defining them as infinite series and presenting "Euler's formula", as well as near-modern abbreviations (sin., cos., tang., cot., sec., and cosec.).

A few functions were common historically, but are now seldom used, such as the chord, the versine (which appeared in the earliest tables), the coversine, the haversine, the exsecant and the excosecant. The list of trigonometric identities shows more relations between these functions.

• crd(θ) = 2 sin(θ/2)
• versin(θ) = 1 − cos(θ) = 2 sin²(θ/2)
• coversin(θ) = 1 − sin(θ) = versin(π/2 − θ))
• haversin(θ) = 1/2versin(θ) = sin²(θ/2)
• exsec(θ) = sec(θ) − 1
• excsc(θ) = exsec(π/2 − θ) = csc(θ) − 1

Etymology

The word sine derives from Latin sinus, meaning "bend; bay", and more specifically "the hanging fold of the upper part of a toga", "the bosom of a garment", which was chosen as the translation of what was interpreted as the Arabic word jaib, meaning "pocket" or "fold" in the twelfth-century translations of works by Al-Battani and al-Khwārizmī into Medieval Latin. The choice was based on a misreading of the Arabic written form j-y-b (جيب), which itself originated as a transliteration from Sanskrit jīvā, which along with its synonym jyā (the standard Sanskrit term for the sine) translates to "bowstring", being in turn adopted from Ancient Greek χορδή "string".

The word tangent comes from Latin tangens meaning "touching", since the line touches the circle of unit radius, whereas secant stems from Latin secans—"cutting"—since the line cuts the circle.

The prefix "co-" (in "cosine", "cotangent", "cosecant") is found in Edmund Gunter's Canon triangulorum (1620), which defines the cosinus as an abbreviation for the sinus complementi (sine of the complementary angle) and proceeds to define the cotangens similarly.

See also

• All Students Take Calculus — a mnemonic for recalling the signs of trigonometric functions in a particular quadrant of a Cartesian plane
• Bhaskara I's sine approximation formula
• Generalized trigonometry
• Generating trigonometric tables
• Hyperbolic function
• List of periodic functions
• List of trigonometric identities
• Polar sine — a generalization to vertex angles
• Proofs of trigonometric identities
• Versine — for several less used trigonometric functions

External links

• Hazewinkel, Michiel, ed. (2001) [1994], "Trigonometric functions", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
• Visionlearning Module on Wave Mathematics
• GonioLab Visualization of the unit circle, trigonometric and hyperbolic functions
• q-Sine Article about the q-analog of sin at MathWorld
• q-Cosine Article about the q-analog of cos at MathWorld

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</math>

のように選ぶことが多い。またこのとき、制限があることを強調するために、Sin⁻¹x, Arcsin x のように頭文字を大文字にした表記がよく用いられる。

複素関数としての三角関数

exp z, cos z, sin z の級数による定義から、オイラーの公式 exp (iz) = cos z + i sin z を導くことができる。この公式から下記の 2 つの等式

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が得られるから、これを連立させて解くことにより、正弦関数・余弦関数の指数関数を用いた表現が可能となる。すなわち、

<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\cos z&={\frac {e^{iz}+e^{-iz}}{2}},\\\sin z&={\frac {e^{iz}-e^{-iz}}{2i}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>z</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>z</mi> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>z</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>z</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\cos z&={\frac {e^{iz}+e^{-iz}}{2}},\\\sin z&={\frac {e^{iz}-e^{-iz}}{2i}}\end{aligned}}}</annotation> </semantics> </math>

が成り立つ。この事実により、級数によらずこの等式をもって複素変数の正弦・余弦関数の定義とすることもある。また、

<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\cos(iz)&={\frac {e^{-z}+e^{z}}{2}}=\cosh z,\\\sin(iz)&={\frac {e^{-z}-e^{z}}{2i}}=i\sinh z\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>z</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mi>cosh</mi> <mo>⁡</mo> <mi>z</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>z</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>i</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>i</mi> <mi>sinh</mi> <mo>⁡</mo> <mi>z</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\cos(iz)&={\frac {e^{-z}+e^{z}}{2}}=\cosh z,\\\sin(iz)&={\frac {e^{-z}-e^{z}}{2i}}=i\sinh z\end{aligned}}}</annotation> </semantics> </math>

が成り立つ。ここで cosh z, sinh z は双曲線関数を表す。この等式は三角関数と双曲線関数の関係式と捉えることもできる。複素数 z を z = x + iy (x, y ∈ R) と表現すると、加法定理より

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が成り立つ。

他の三角関数は csc z = 1 / sin z, sec z = 1 / cos z, tan z = sin z / cos z, cot z = cos z / sin z によって定義できる。

• cos(x + iy) の実部のグラフ

• cos(x + iy) の虚部のグラフ

• sin(x + iy) の実部のグラフ

• sin(x + iy) の虚部のグラフ

球面三角法

詳細は「球面三角法」を参照

球面の三角形 ABC の内角を a, b, c, 各頂点の対辺に関する球の中心角を α, β, γ とするとき、次のような関係が成立する。余弦公式や正弦余弦公式は式の対称性により各記号を入れ替えたものも成立する。

正弦公式

sin a : sin b : sin c = sin α : sin β : sin γ

余弦公式

cos a = −cos b cos c + sin b sin c cos α

余弦公式

cos α = cos β cos γ + sin β sin γ cos a

正弦余弦公式

sin a cos β = cos b sin c − sin b cos c cos α

出典

<tbody></tbody>

参考文献

• Maor, Eli (1998). Trigonometric Delights. Princeton University Press. ISBN 978069105754-5. 
• 志賀, 浩二 『数の大航海―対数の誕生と広がり』 日本評論社、1999年7月。ISBN 978-4-535-78289-1。
• 高瀬, 正仁 『古典的難問に学ぶ微分積分』 共立出版、2013年7月。ISBN 978-4-320-11041-0。
• Vinogradov, Ivan Matveyevich (2004-09-10). The Method of Trigonometrical Sums in the Theory of Numbers (revised ed.). Dover. ISBN 978-048643878-8. 
• 黒川, 信重、小山, 信也 『多重三角関数論講義』 日本評論社、2010年11月8日。ISBN 978-4-535-785557。
• 杉浦, 光夫 『解析入門I』 東京大学出版会〈基礎数学2〉、1980年。ISBN 978-4-13-062005-5。
• 黒田, 成俊 『微分積分』 共立出版〈共立講座21世紀の数学 第1巻〉、2002年。ISBN 978-4320015531。
• 高木, 貞治 『定本 解析概論』 岩波書店、2010年、改訂第3版。ISBN 978-4000052092。
• 小平, 邦彦 『解析入門I』 岩波書店、2003年、軽装版。ISBN 978-4000051927。

関連項目

• 正弦定理
• 余弦定理
• 正接定理
• 球面三角法
• コサイン4乗則
• ベクトルのなす角 - cos 関数を用いて表現される。
• ドット積
• クロス積
• ベッセル関数
• sinc関数
• 指数関数
• 双曲線関数
• オイラーの公式
• 円_(数学) - 正円の三角関数との関係
• ベジエ曲線 - 三角関数のベジエ曲線による近似
• テイラー展開 - コンピュータ上での三角関数の実装に使用

外部リンク

• 三角比の近似値表
• Weisstein, Eric W. "Trigonometric Functions". MathWorld（英語）. CS1 maint: Multiple names: authors list

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"Cosine" redirects here. For the similarity measure, see Cosine similarity.

Trigonometry
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Basis of trigonometry: if two right triangles have equal acute angles, they are similar, so their side lengths are proportional. Proportionality constants are written within the image: sin θ, cos θ, tan θ, where θ is the common measure of five acute angles.

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are functions of an angle. They relate the angles of a triangle to the lengths of its sides. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications.

The most familiar trigonometric functions are the sine, cosine, and tangent. In the context of the standard unit circle (a circle with radius 1 unit), where a triangle is formed by a ray starting at the origin and making some angle with the x-axis, the sine of the angle gives the y-component (the opposite to the angle or the rise) of the triangle, the cosine gives the x-component (the adjacent of the angle or the run), and the tangent function gives the slope (y-component divided by the x-component). For angles less than a right angle, trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle, and their values can be found in the lengths of various line segments around a unit circle. Modern definitions express trigonometric functions as infinite series or as solutions of certain differential equations, allowing the extension of the arguments to the whole number line and to the complex numbers.

Trigonometric functions have a wide range of uses including computing unknown lengths and angles in triangles (often right triangles). In this use, trigonometric functions are used, for instance, in navigation, engineering, and physics. A common use in elementary physics is resolving a vector into Cartesian coordinates. The sine and cosine functions are also commonly used to model periodic function phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations through the year.

In modern usage, there are six basic trigonometric functions, tabulated here with equations that relate them to one another. Especially with the last four, these relations are often taken as the definitions of those functions, but one can define them equally well geometrically, or by other means, and then derive these relations.

Right-angled triangle definitions

Top: Trigonometric function sin θ for selected angles θ, π − θ, π + θ, and 2π − θ in the four quadrants.
Bottom: Graph of sine function versus angle. Angles from the top panel are identified.

Plot of the six trigonometric functions and the unit circle for an angle of 0.7 radians.

The notion that there should be some standard correspondence between the lengths of the sides of a triangle and the angles of the triangle comes as soon as one recognizes that similar triangles maintain the same ratios between their sides. That is, for any similar triangle the ratio of the hypotenuse (for example) and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides. It is these ratios that the trigonometric functions express.

To define the trigonometric functions for the angle A, start with any right triangle that contains the angle A. The three sides of the triangle are named as follows:

• The hypotenuse is the side opposite the right angle, in this case side h. The hypotenuse is always the longest side of a right-angled triangle.
• The opposite side is the side opposite to the angle we are interested in (angle A), in this case side a.
• The adjacent side is the side having both the angles of interest (angle A and right-angle C), in this case side b.

In ordinary Euclidean geometry, according to the triangle postulate, the inside angles of every triangle total 180° (π radians). Therefore, in a right-angled triangle, the two non-right angles total 90° (π/2 radians), so each of these angles must be in the range of (0, π/2) as expressed in interval notation. The following definitions apply to angles in this (0, π/2) range. They can be extended to the full set of real arguments by using the unit circle, or by requiring certain symmetries and that they be periodic functions. For example, the figure shows sin(θ) for angles θ, π − θ, π + θ, and 2π − θ depicted on the unit circle (top) and as a graph (bottom). The value of the sine repeats itself apart from sign in all four quadrants, and if the range of θ is extended to additional rotations, this behavior repeats periodically with a period 2π.

The trigonometric functions are summarized in the following table and described in more detail below. The angle θ is the angle between the hypotenuse and the adjacent line – the angle at A in the accompanying diagram.

Function	Abbreviation	Description	Identities (using radians)
sine	sin	opposite/hypotenuse	$$ sin ⁡ θ = cos ⁡ ( π 2 − θ ) = 1 csc ⁡ θ {\displaystyle \sin \theta =\cos \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\csc \theta }}}
cosine	cos	adjacent/hypotenuse	$$ cos ⁡ θ = sin ⁡ ( π 2 − θ ) = 1 sec ⁡ θ {\displaystyle \cos \theta =\sin \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\sec \theta }}\,}
tangent	tan (or tg)	opposite/adjacent	$$ tan ⁡ θ = sin ⁡ θ cos ⁡ θ = cot ⁡ ( π 2 − θ ) = 1 cot ⁡ θ {\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}=\cot \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cot \theta }}}
cotangent	cot (or cotan or cotg or ctg or ctn)	adjacent/opposite	$$ cot ⁡ θ = cos ⁡ θ sin ⁡ θ = tan ⁡ ( π 2 − θ ) = 1 tan ⁡ θ {\displaystyle \cot \theta ={\frac {\cos \theta }{\sin \theta }}=\tan \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\tan \theta }}}
secant	sec	hypotenuse/adjacent	$$ sec ⁡ θ = csc ⁡ ( π 2 − θ ) = 1 cos ⁡ θ {\displaystyle \sec \theta =\csc \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\cos \theta }}}
cosecant	csc (or cosec)	hypotenuse/opposite	$$ csc ⁡ θ = sec ⁡ ( π 2 − θ ) = 1 sin ⁡ θ {\displaystyle \csc \theta =\sec \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\sin \theta }}}

Sine, cosine and tangent

The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. The word comes from the Latin sinus for gulf or bay, since, given a unit circle, it is the side of the triangle on which the angle opens. In our case:

$\sin A={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}$

An illustration of the relationship between sine and its out-of-phase complement, cosine. Cosine is identical, but π/2 radians out of phase to the left; so cos A = sin(A + π/2).

The cosine (sine complement, Latin: cosinus, sinus complementi) of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse, so called because it is the sine of the complementary or co-angle, the other non-right angle. Because the angle sum of a triangle is π radians, the co-angle B is equal to π/2 − A; so cos A = sin B = sin(π/2 − A). In our case:

$\cos A={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}$

The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side, so called because it can be represented as a line segment tangent to the circle, i.e. the line that touches the circle, from Latin linea tangens or touching line (cf. tangere, to touch). In our case:

$\tan A={\frac {\textrm {opposite}}{\textrm {adjacent}}}$

Tangent may also be represented in terms of sine and cosine. That is:

$\tan A={\frac {\sin A}{\cos A}}={\frac {\frac {\textrm {opposite}}{\textrm {hypotenuse}}}{\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}}={\frac {\textrm {opposite}}{\textrm {adjacent}}}$

These ratios do not depend on the size of the particular right triangle chosen, as long as the focus angle is equal, since all such triangles are similar.

The acronyms "SOH-CAH-TOA" ("soak-a-toe", "sock-a-toa", "so-kah-toa") and "OHSAHCOAT" are commonly used trigonometric mnemonics for these ratios.

Secant, cosecant and cotangent

The remaining three functions are best defined using the three functions above and can be considered their reciprocals.

The secant of an angle is the reciprocal of its cosine, that is, the ratio of the length of the hypotenuse to the length of the adjacent side, so called because it represents the secant line that cuts the circle (from Latin: secare, to cut):

$\sec A={\frac {1}{\cos A}}={\frac {\textrm {hypotenuse}}{\textrm {adjacent}}}={\frac {h}{b}}.$

The cosecant (secant complement, Latin: cosecans, secans complementi) of an angle is the reciprocal of its sine, that is, the ratio of the length of the hypotenuse to the length of the opposite side, so called because it is the secant of the complementary or co-angle:

$\csc A={\frac {1}{\sin A}}={\frac {\textrm {hypotenuse}}{\textrm {opposite}}}={\frac {h}{a}}.$

The cotangent (tangent complement, Latin: cotangens, tangens complementi) of an angle is the reciprocal of its tangent, that is, the ratio of the length of the adjacent side to the length of the opposite side, so called because it is the tangent of the complementary or co-angle:

$\cot A={\frac {1}{\tan A}}={\frac {\textrm {adjacent}}{\textrm {opposite}}}={\frac {b}{a}}.$

Mnemonics

Equivalent to the right-triangle definitions, the trigonometric functions can also be defined in terms of the rise, run, and slope of a line segment relative to horizontal. The slope is commonly taught as "rise over run" or rise/run. The three main trigonometric functions are commonly taught in the order sine, cosine and tangent. With a line segment length of 1 (as in a unit circle), the following mnemonic devices show the correspondence of definitions:

1. "Sine is first, rise is first" meaning that Sine takes the angle of the line segment and tells its vertical rise when the length of the line is 1.
2. "Cosine is second, run is second" meaning that Cosine takes the angle of the line segment and tells its horizontal run when the length of the line is 1.
3. "Tangent combines the rise and run" meaning that Tangent takes the angle of the line segment and tells its slope, or alternatively, tells the vertical rise when the line segment's horizontal run is 1.

This shows the main use of tangent and arctangent: converting between the two ways of telling the slant of a line, i.e. angles and slopes. (The arctangent or "inverse tangent" is not to be confused with the cotangent, which is cosine divided by sine.)

While the length of the line segment makes no difference for the slope (the slope does not depend on the length of the slanted line), it does affect rise and run. To adjust and find the actual rise and run when the line does not have a length of 1, just multiply the sine and cosine by the line length. For instance, if the line segment has length 5, the run at an angle of 7° is 5cos(7°).

Unit-circle definitions

In this illustration, the six trigonometric functions of an arbitrary angle θ are represented as Cartesian coordinates of points related to the unit circle. The ordinates of A, B and D are sin θ, tan θ and csc θ, respectively, while the abscissas of A, C and E are cos θ, cot θ and sec θ, respectively.

Signs of trigonometric functions in each quadrant. The mnemonic "all science teachers (are) crazy" lists the functions which are positive from quadrants I to IV. This is a variation on the mnemonic "All Students Take Calculus".

The six trigonometric functions can be defined as coordinate values of points on the Euclidean plane that are related to the unit circle, which is the circle of radius one centered at the origin O of this coordinate system. While right-angled triangle definitions permit the definition of the trigonometric functions for angles between 0 and ${\tfrac {\pi }{2}}$ radian (90°), the unit circle definitions allow to extend the domain of the trigonometric functions to all positive and negative real numbers.

Rotating a ray from the direction of the positive half of the x-axis by an angle θ (counterclockwise for $$

 
   
     
       θ
       >
       0
       ,
     
   
   {\displaystyle \theta >0,}
 

and clockwise for

 
   
     
       θ
       <
       0
     
   
   {\displaystyle \theta <0}
 

) yields intersection points of this ray (see the figure) with the unit circle: $$

 
   
     
       
         A
       
       =
       (
       
         x
         
           
             A
           
         
       
       ,
       
         y
         
           
             A
           
         
       
       )
     
   
   {\displaystyle \mathrm {A} =(x_{\mathrm {A} },y_{\mathrm {A} })}
 

, and, by extending the ray to a line if necessary, with the line $$

 
   
     
       
         “
       
       x
       =
       1
       
         ”
       
       :
       
       
         B
       
       =
       (
       
         x
         
           
             B
           
         
       
       ,
       
         y
         
           
             B
           
         
       
       )
       ,
     
   
   {\displaystyle {\text{“}}x=1{\text{”}}:\;\mathrm {B} =(x_{\mathrm {B} },y_{\mathrm {B} }),}
 

and with the line $$

 
   
     
       
         “
       
       y
       =
       1
       
         ”
       
       :
       
       
         C
       
       =
       (
       
         x
         
           
             C
           
         
       
       ,
       
         y
         
           
             C
           
         
       
       )
       .
     
   
   {\displaystyle {\text{“}}y=1{\text{”}}:\;\mathrm {C} =(x_{\mathrm {C} },y_{\mathrm {C} }).}
 

The tangent line to the unit circle in point A, which is orthogonal to this ray, intersects the y- and x-axis in points $$

 
   
     
       
         D
       
       =
       (
       0
       ,
       
         y
         
           
             D
           
         
       
       )
     
   
   {\displaystyle \mathrm {D} =(0,y_{\mathrm {D} })}
 

and $$

 
   
     
       
         E
       
       =
       (
       
         x
         
           
             E
           
         
       
       ,
       0
       )
     
   
   {\displaystyle \mathrm {E} =(x_{\mathrm {E} },0)}
 

. The coordinate values of these points give all the existing values of the trigonometric functions for arbitrary real values of θ in the following manner.

The trigonometric functions cos and sin are defined, respectively, as the x- and y-coordinate values of point A, i.e.,

$\cos(\theta )=x_{\mathrm {A} }\quad$ and $\quad \sin(\theta )=y_{\mathrm {A} }.$

In the range $0\leq \theta \leq \pi /2$ this definition coincides with the right-angled triangle definition by taking the right-angled triangle to have the unit radius OA as hypotenuse, and since for all points $\mathrm {P} =(x,y)$ on the unit circle the equation $x^{2}+y^{2}=1$ holds, this definition of cosine and sine also satisfies the Pythagorean identity

$\cos ^{2}\theta +\sin ^{2}\theta =1.$

The other trigonometric functions can be found along the unit circle as

$\tan(\theta )=y_{\mathrm {B} }\quad$ and $\quad \cot(\theta )=x_{\mathrm {C} },$

$\csc(\theta )\ =y_{\mathrm {D} }\quad$ and $\quad \sec(\theta )=x_{\mathrm {E} }.$

By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is

$\tan \theta ={\frac {\sin \theta }{\cos \theta }},\quad \cot \theta ={\frac {\cos \theta }{\sin \theta }},\quad \sec \theta ={\frac {1}{\cos \theta }},\quad \csc \theta ={\frac {1}{\sin \theta }}.$

Trigonometric functions: Sine, Cosine, Tangent, Cosecant (dotted), Secant (dotted), Cotangent (dotted)

As a rotation of an angle of $\pm 2\pi$ does not change the position or size of a shape, the points A, B, C, D, and E are the same for two angles whose difference is an integer multiple of $2\pi$. Thus trigonometric functions are periodic functions with period $2\pi$. That is, the equalities

$\sin \theta =\sin \left(\theta +2k\pi \right)\quad$ and $\quad \cos \theta =\cos \left(\theta +2k\pi \right)$

hold for any angle θ and any integer k. The same is true for the four other trigonometric functions. Observing the sign and the monotonicity of the functions sine, cosine, cosecant, and secant in the four quadrants, shows that 2π is the smallest value for which they are periodic, i.e., 2π is the fundamental period of these functions. However, already after a rotation by an angle $\pi$ the points B and C return to their original position, so that the tangent function and the cotangent function have a fundamental period of π. That is, the equalities

$\tan \theta =\tan(\theta +k\pi )\quad$ and $\quad \cot \theta =\cot(\theta +k\pi )$

hold for any angle θ and any integer k.

Algebraic values

The unit circle, with some points labeled with their cosine and sine (in this order), and the corresponding angles in radians and degrees.

The algebraic expressions for sin(0°), sin(30°), sin(45°), sin(60°) and sin(90°) are

$0,\quad {\frac {1}{2}},\quad {\frac {\sqrt {2}}{2}},\quad {\frac {\sqrt {3}}{2}},\quad 1,\,\!$

respectively. Writing the numerators as square roots of consecutive natural numbers (${\tfrac {\sqrt {0}}{2}},{\tfrac {\sqrt {1}}{2}},{\tfrac {\sqrt {2}}{2}},{\tfrac {\sqrt {3}}{2}},{\tfrac {\sqrt {4}}{2}}$) provides an easy way to remember the values. Such simple expressions generally do not exist for other angles which are rational multiples of a straight angle.

For an angle which, measured in degrees, is a multiple of three, the sine and the cosine may be expressed in terms of square roots, as shown below. These values of the sine and the cosine may thus be constructed by ruler and compass.

For an angle of an integer number of degrees, the sine and the cosine may be expressed in terms of square roots and the cube root of a non-real complex number. Galois theory allows proof that, if the angle is not a multiple of 3°, non-real cube roots are unavoidable.

For an angle which, measured in degrees, is a rational number, the sine and the cosine are algebraic numbers, which may be expressed in terms of nth roots. This results from the fact that the Galois groups of the cyclotomic polynomials are cyclic.

For an angle which, measured in degrees, is not a rational number, then either the angle or both the sine and the cosine are transcendental numbers. This is a corollary of Baker's theorem, proved in 1966.

Explicit values

Main article: Trigonometric constants expressed in real radicals

Algebraic expressions for 15°, 18°, 36°, 54°, 72° and 75° are as follows:

$\sin 15^{\circ }=\cos 75^{\circ }={\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}\,\!$

$\sin 18^{\circ }=\cos 72^{\circ }={\frac {{\sqrt {5}}-1}{4}}$

$\sin 36^{\circ }=\cos 54^{\circ }={\frac {\sqrt {10-2{\sqrt {5}}}}{4}}$

$\sin 54^{\circ }=\cos 36^{\circ }={\frac {{\sqrt {5}}+1}{4}}\,\!$

$\sin 72^{\circ }=\cos 18^{\circ }={\frac {\sqrt {10+2{\sqrt {5}}}}{4}}$

$\sin 75^{\circ }=\cos 15^{\circ }={\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}.\,$

From these, the algebraic expressions for all multiples of 3° can be computed. For example:

$\sin 3^{\circ }=\cos 87^{\circ }={\frac {2\left(1-{\sqrt {3}}\right){\sqrt {5+{\sqrt {5}}}}+\left(1+{\sqrt {3}}\right)\left({\sqrt {10}}-{\sqrt {2}}\right)}{16}}\,\!$

$\sin 6^{\circ }=\cos 84^{\circ }={\frac {{\sqrt {30-6{\sqrt {5}}}}-{\sqrt {5}}-1}{8}}\,\!$

$\sin 9^{\circ }=\cos 81^{\circ }={\frac {{\sqrt {10}}+{\sqrt {2}}-2{\sqrt {5-{\sqrt {5}}}}}{8}}\,\!$

$\sin 84^{\circ }=\cos 6^{\circ }={\frac {{\sqrt {10-2{\sqrt {5}}}}+{\sqrt {3}}+{\sqrt {15}}}{8}}$

$\sin 87^{\circ }=\cos 3^{\circ }={\frac {2\left(1+{\sqrt {3}}\right){\sqrt {5+{\sqrt {5}}}}-\left(1-{\sqrt {3}}\right)\left({\sqrt {10}}-{\sqrt {2}}\right)}{16}}.$

Algebraic expressions can be deduced for other angles of an integer number of degrees, for example,

$\sin 1^{\circ }={\frac {{\sqrt[{3}]{z}}-{\dfrac {1}{\sqrt[{3}]{z}}}}{2i}},$

where z = a + ib, and a and b are the above algebraic expressions for, respectively, cos 3° and sin 3°, and the principal cube root (that is, the cube root with the largest real part) is to be taken.

Series definitions

The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full cycle centered on the origin.

Animation for the approximation of cosine via Taylor polynomials.

$$

 
   
     
       cos
       ⁡
       (
       x
       )
     
   
   {\displaystyle \cos(x)}
 

together with the first Taylor polynomials $$

 
   
     
       
         p
         
           n
         
       
       (
       x
       )
       =
       
         ∑
         
           k
           =
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           n
         
       
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       −
       1
       
         )
         
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             x
             
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               k
             
           
           
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             k
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   {\displaystyle p_{n}(x)=\sum _{k=0}^{n}(-1)^{k}{\frac {x^{2k}}{(2k)!}}}
 

Trigonometric functions are analytic functions. Using only geometry and properties of limits, it can be shown that the derivative of sine is cosine and the derivative of cosine is the negative of sine. One can then use the theory of Taylor series to show that the following identities hold for all real numbers x. Here, and generally in calculus, all angles are measured in radians.

${\begin{aligned}\sin x&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots \\&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}\\\end{aligned}}$

${\begin{aligned}\cos x&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots \\&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}\end{aligned}}$

The infinite series appearing in these identities are convergent in the whole complex plane and are often taken as the definitions of the sine and cosine functions of a complex variable. Another standard (and equivalent) definition of the sine and the cosine as functions of a complex variable is through their differential equation, below.

Other series can be found. For the following trigonometric functions:

U_n is the nth up/down number,

B_n is the nth Bernoulli number, and

E_n (below) is the nth Euler number.

${\begin{aligned}\tan x&{}=\sum _{n=0}^{\infty }{\frac {U_{2n+1}x^{2n+1}}{(2n+1)!}}\\&{}=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}2^{2n}(2^{2n}-1)B_{2n}x^{2n-1}}{(2n)!}}\\&{}=x+{\frac {1}{3}}x^{3}+{\frac {2}{15}}x^{5}+{\frac {17}{315}}x^{7}+\cdots ,\qquad {\text{for }}|x|<{\frac {\pi }{2}}.\end{aligned}}$

${\begin{aligned}\csc x&{}=\sum _{n=0}^{\infty }{\frac {(-1)^{n+1}2(2^{2n-1}-1)B_{2n}x^{2n-1}}{(2n)!}}\\&{}=x^{-1}+{\frac {1}{6}}x+{\frac {7}{360}}x^{3}+{\frac {31}{15120}}x^{5}+\cdots ,\qquad {\text{for }}0<|x|<\pi .\end{aligned}}$

${\begin{aligned}\sec x&{}=\sum _{n=0}^{\infty }{\frac {U_{2n}x^{2n}}{(2n)!}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}E_{2n}x^{2n}}{(2n)!}}\\&{}=1+{\frac {1}{2}}x^{2}+{\frac {5}{24}}x^{4}+{\frac {61}{720}}x^{6}+\cdots ,\qquad {\text{for }}|x|<{\frac {\pi }{2}}.\end{aligned}}$

${\begin{aligned}\cot x&{}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}2^{2n}B_{2n}x^{2n-1}}{(2n)!}}\\&{}=x^{-1}-{\frac {1}{3}}x-{\frac {1}{45}}x^{3}-{\frac {2}{945}}x^{5}-\cdots ,\qquad {\text{for }}0<|x|<\pi .\end{aligned}}$

When the series for the tangent and secant functions are expressed in a form in which the denominators are the corresponding factorials, the numerators, called the "tangent numbers" and "secant numbers" respectively, have a combinatorial interpretation: they enumerate alternating permutations of finite sets, of odd cardinality for the tangent series and even cardinality for the secant series. The series itself can be found by a power series solution of the aforementioned differential equation.

From a theorem in complex analysis, there is a unique analytic continuation of this real function to the domain of complex numbers. They have the same Taylor series, and so the trigonometric functions are defined on the complex numbers using the Taylor series above.

There is a series representation as partial fraction expansion where just translated reciprocal functions are summed up, such that the poles of the cotangent function and the reciprocal functions match:

$\pi \cdot \cot(\pi x)=\lim _{N\to \infty }\sum _{n=-N}^{N}{\frac {1}{x+n}}.$

This identity can be proven with the Herglotz trick. Combining the (–n)th with the nth term lead to absolutely convergent series:

$\pi \cdot \cot(\pi x)={\frac {1}{x}}+\sum _{n=1}^{\infty }{\frac {2x}{x^{2}-n^{2}}}\ ,\quad {\frac {\pi }{\sin(\pi x)}}={\frac {1}{x}}+\sum _{n=1}^{\infty }{\frac {(-1)^{n}\,2x}{x^{2}-n^{2}}}.$

Relationship to exponential function and complex numbers

Euler's formula illustrated with the three dimensional helix, starting with the 2D orthogonal components of the unit circle, sine and cosine (using θ = t).

$$

 
   
     
       cos
       ⁡
       (
       θ
       )
     
   
   {\displaystyle \cos(\theta )}
 

and $$

 
   
     
       sin
       ⁡
       (
       θ
       )
     
   
   {\displaystyle \sin(\theta )}
 

are the real and imaginary part of $$

 
   
     
       
         
           e
         
         
           
             i
           
           θ
         
       
     
   
   {\displaystyle \mathrm {e} ^{\mathrm {i} \theta }}
 

respectively.

It can be shown from the series definitions that the sine and cosine functions are respectively the imaginary and real parts of the exponential function of a purely imaginary argument. That is, if x is real, we have

$\cos x=\operatorname {Re} (e^{ix})\,,\qquad \sin x=\operatorname {Im} (e^{ix})\,,$

and

$e^{ix}=\cos x+i\sin x\,.$

The latter identity, although primarily established for real x, remains valid for every complex x, and is called Euler's formula.

Euler's formula can be used to derive most trigonometric identities from the properties of the exponential function, by writing sine and cosine as:

$\sin x={\frac {e^{ix}-e^{-ix}}{2i}}\,,\qquad \cos x={\frac {e^{ix}+e^{-ix}}{2}}\,.$

It is also sometimes useful to express the complex sine and cosine functions in terms of the real and imaginary parts of their arguments.

${\begin{aligned}\sin(x+iy)&=\sin x\cosh y+i\cos x\sinh y\,,\\\cos(x+iy)&=\cos x\cosh y-i\sin x\sinh y\,.\end{aligned}}$

This exhibits a deep relationship between the complex sine and cosine functions and their real (sin, cos) and hyperbolic real (sinh, cosh) counterparts.

Complex graphs

In the following graphs the domain is the complex plane pictured with domain coloring, and the range values are indicated at each point by color. Brightness indicates the size (absolute value) of the range value, with black being zero. Hue varies with argument, or angle, measured from the positive real axis.

Trigonometric functions in the complex plane

$$ sin ⁡ z {\displaystyle \sin z\,}	$$ cos ⁡ z {\displaystyle \cos z\,}	$$ tan ⁡ z {\displaystyle \tan z\,}	$$ cot ⁡ z {\displaystyle \cot z\,}	$$ sec ⁡ z {\displaystyle \sec z\,}	$$ csc ⁡ z {\displaystyle \csc z\,}

Definitions via differential equations

Both the sine and cosine functions satisfy the linear differential equation:

$y''=-y\,.$

That is to say, each is the additive inverse of its own second derivative. Within the 2-dimensional function space V consisting of all solutions of this equation,

• the sine function is the unique solution satisfying the initial condition $\left(y'(0),y(0)\right)=(1,0)\,$ and
• the cosine function is the unique solution satisfying the initial condition $\left(y'(0),y(0)\right)=(0,1)\,$.

Since the sine and cosine functions are linearly independent, together they form a basis of V. This method of defining the sine and cosine functions is essentially equivalent to using Euler's formula. (See linear differential equation.) It turns out that this differential equation can be used not only to define the sine and cosine functions but also to prove the trigonometric identities for the sine and cosine functions.

Further, the observation that sine and cosine satisfies y″ = −y means that they are eigenfunctions of the second-derivative operator.

The tangent function is the unique solution of the nonlinear differential equation

$y'=1+y^{2}\,$

satisfying the initial condition y(0) = 0. There is a very interesting visual proof that the tangent function satisfies this differential equation.

The significance of radians

Radians specify an angle by measuring the length around the path of the unit circle and constitute a special argument to the sine and cosine functions. In particular, only sines and cosines that map radians to ratios satisfy the differential equations that classically describe them. If an argument to sine or cosine in radians is scaled by frequency,

$f(x)=\sin(kx),\,$

then the derivatives will scale by amplitude.

$f'(x)=k\cos(kx).\,$

Here, k is a constant that represents a mapping between units. If x is in degrees, then

$k={\frac {\pi }{180^{\circ }}}.$

This means that the second derivative of a sine in degrees does not satisfy the differential equation

$y''=-y\,$

but rather

$y''=-k^{2}y.\,$

The cosine's second derivative behaves similarly.

This means that these sines and cosines are different functions, and that the fourth derivative of sine will be sine again only if the argument is in radians.

Identities

Main articles: List of trigonometric identities and Proofs of trigonometric identities

Many identities interrelate the trigonometric functions. Among the most frequently used is the Pythagorean identity, which states that for any angle, the square of the sine plus the square of the cosine is 1. This is easy to see by studying a right triangle of hypotenuse 1 and applying the Pythagorean theorem. In symbolic form, the Pythagorean identity is written

$\sin ^{2}x+\cos ^{2}x=1$

which is standard shorthand notation for

$(\sin x)^{2}+(\cos x)^{2}=1.$

Other key relationships are the sum and difference formulas, which give the sine and cosine of the sum and difference of two angles in terms of sines and cosines of the angles themselves. These can be derived geometrically, using arguments that date to Ptolemy. One can also produce them algebraically using Euler's formula.

Sum

${\begin{aligned}\sin \left(x+y\right)&=\sin x\cos y+\cos x\sin y,\\\cos \left(x+y\right)&=\cos x\cos y-\sin x\sin y,\end{aligned}}$

Difference

${\begin{aligned}\sin \left(x-y\right)&=\sin x\cos y-\cos x\sin y,\\\cos \left(x-y\right)&=\cos x\cos y+\sin x\sin y.\end{aligned}}$

These in turn lead to the following three-angle formulae:

${\begin{aligned}\sin \left(x+y+z\right)&=\sin x\cos y\cos z+\sin y\cos z\cos x+\sin z\cos y\cos x-\sin x\sin y\sin z,\\\cos \left(x+y+z\right)&=\cos x\cos y\cos z-\cos x\sin y\sin z-\cos y\sin x\sin z-\cos z\sin x\sin y,\end{aligned}}$

When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulae.

${\begin{aligned}\sin \left(2x\right)&=2\sin x\cos x,\\\cos \left(2x\right)&=\cos ^{2}x-\sin ^{2}x=2\cos ^{2}x-1=1-2\sin ^{2}x.\end{aligned}}$

When three angles are equal, the three-angle formulae simplify to

${\begin{aligned}\sin(3x)&=3\sin x-4\sin ^{3}x.\\\cos(3x)&=4\cos ^{3}x-3\cos x.\end{aligned}}$

These identities can also be used to derive the product-to-sum identities that were used in antiquity to transform the product of two numbers into a sum of numbers and greatly speed operations, much like the logarithm function.

Calculus

For integrals and derivatives of trigonometric functions, see the relevant sections of Differentiation of trigonometric functions, Lists of integrals and List of integrals of trigonometric functions. Below is the list of the derivatives and integrals of the six basic trigonometric functions. The number C is a constant of integration.

${\begin{array}{|r|rr|}f(x)&f'(x)&\int f(x)\,dx\\\hline \sin x&\cos x&-\cos x+C\\\cos x&-\sin x&\sin x+C\\\tan x&\sec ^{2}x=1+\tan ^{2}x&-\ln \left|\cos x\right|+C\\\cot x&-\csc ^{2}x=-(1+\cot ^{2}x)&\ln \left|\sin x\right|+C\\\sec x&\sec x\tan x&\ln \left|\sec x+\tan x\right|+C\\\csc x&-\csc x\cot x&-\ln \left|\csc x+\cot x\right|+C\end{array}}$

Definitions using functional equations

In mathematical analysis, one can define the trigonometric functions using functional equations based on properties like the difference formula. Taking as given these formulas, one can prove that only two continuous functions satisfy those conditions. Formally, there exists exactly one pair of continuous functions—sin and cos—such that for all real numbers x and y, the following equation holds:

$\cos(x-y)=\cos x\cos y+\sin x\sin y\,$

with the added condition that

$0<x\cos x<\sin x<x\quad {\text{ for }}\quad 0<x<1.\,$

This may also be used for extending sine and cosine to the complex numbers. Other functional equations are also possible for defining trigonometric functions.

Computation

The computation of trigonometric functions is a complicated subject, which can today be avoided by most people because of the widespread availability of computers and scientific calculators that provide built-in trigonometric functions for any angle. This section, however, describes details of their computation in three important contexts: the historical use of trigonometric tables, the modern techniques used by computers, and a few "important" angles where simple exact values are easily found.

The first step in computing any trigonometric function is range reduction—reducing the given angle to a "reduced angle" inside a small range of angles, say 0 to π/2, using the periodicity and symmetries of the trigonometric functions.

Main article: Generating trigonometric tables

Prior to computers, people typically evaluated trigonometric functions by interpolating from a detailed table of their values, calculated to many significant figures. Such tables have been available for as long as trigonometric functions have been described (see History below), and were typically generated by repeated application of the half-angle and angle-addition identities starting from a known value (such as sin(π/2) = 1).

Modern computers use a variety of techniques. One common method, especially on higher-end processors with floating point units, is to combine a polynomial or rational approximation (such as Chebyshev approximation, best uniform approximation, and Padé approximation, and typically for higher or variable precisions, Taylor and Laurent series) with range reduction and a table lookup—they first look up the closest angle in a small table, and then use the polynomial to compute the correction. Devices that lack hardware multipliers often use an algorithm called CORDIC (as well as related techniques), which uses only addition, subtraction, bitshift, and table lookup. These methods are commonly implemented in hardware floating-point units for performance reasons.

For very high precision calculations, when series expansion convergence becomes too slow, trigonometric functions can be approximated by the arithmetic-geometric mean, which itself approximates the trigonometric function by the (complex) elliptic integral.

Main article: Exact trigonometric constants

Finally, for some simple angles, the values can be easily computed by hand using the Pythagorean theorem, as in the following examples. For example, the sine, cosine and tangent of any integer multiple of π/60 radians (3°) can be found exactly by hand.

Consider a right triangle where the two other angles are equal, and therefore are both π/4 radians (45°). Then the length of side b and the length of side a are equal; we can choose a = b = 1. The values of sine, cosine and tangent of an angle of π/4 radians (45°) can then be found using the Pythagorean theorem:

$c={\sqrt {a^{2}+b^{2}}}={\sqrt {2}}\,.$

Therefore:

$\sin {\frac {\pi }{4}}=\sin 45^{\circ }=\cos {\frac {\pi }{4}}=\cos 45^{\circ }={1 \over {\sqrt {2}}}={{\sqrt {2}} \over 2},\,$

$\tan {\frac {\pi }{4}}=\tan 45^{\circ }={{\sin {\frac {\pi }{4}}} \over {\cos {\frac {\pi }{4}}}}={1 \over {\sqrt {2}}}\cdot {{\sqrt {2}} \over 1}={{\sqrt {2}} \over {\sqrt {2}}}=1.\,$

Computing trigonometric functions from an equilateral triangle

To determine the trigonometric functions for angles of π/3 radians (60°) and π/6 radians (30°), we start with an equilateral triangle of side length 1. All its angles are π/3 radians (60°). By dividing it into two, we obtain a right triangle with π/6 radians (30°) and π/3 radians (60°) angles. For this triangle, the shortest side is 1/2, the next largest side is √3/2 and the hypotenuse is 1. This yields:

$\sin {\frac {\pi }{6}}=\sin 30^{\circ }=\cos {\frac {\pi }{3}}=\cos 60^{\circ }={1 \over 2}\,,$

$\cos {\frac {\pi }{6}}=\cos 30^{\circ }=\sin {\frac {\pi }{3}}=\sin 60^{\circ }={{\sqrt {3}} \over 2}\,,$

$\tan {\frac {\pi }{6}}=\tan 30^{\circ }=\cot {\frac {\pi }{3}}=\cot 60^{\circ }={1 \over {\sqrt {3}}}={{\sqrt {3}} \over 3}\,.$

Special values in trigonometric functions

There are some commonly used special values in trigonometric functions, as shown in the following table.

${\begin{array}{|c|cccccccc|}{\begin{matrix}{\text{Radian}}\\{\text{Degree}}\end{matrix}}&{\begin{matrix}0\^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{12}}\\15^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{8}}\\22.5^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{6}}\\30^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{4}}\\45^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{3}}\\60^{\circ }\end{matrix}}&{\begin{matrix}{\frac {5\pi }{12}}\\75^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{2}}\\90^{\circ }\end{matrix}}\\\hline \sin &0&{\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}&{\frac {\sqrt {2-{\sqrt {2}}}}{2}}&{\frac {1}{2}}&{\frac {\sqrt {2}}{2}}&{\frac {\sqrt {3}}{2}}&{\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}&1\\\cos &1&{\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}&{\frac {\sqrt {2+{\sqrt {2}}}}{2}}&{\frac {\sqrt {3}}{2}}&{\frac {\sqrt {2}}{2}}&{\frac {1}{2}}&{\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}&0\\\tan &0&2-{\sqrt {3}}&{\sqrt {2}}-1&{\frac {\sqrt {3}}{3}}&1&{\sqrt {3}}&2+{\sqrt {3}}&\infty \\\cot &\infty &2+{\sqrt {3}}&{\sqrt {2}}+1&{\sqrt {3}}&1&{\frac {\sqrt {3}}{3}}&2-{\sqrt {3}}&0\\\sec &1&{\sqrt {6}}-{\sqrt {2}}&{\sqrt {2}}{\sqrt {2-{\sqrt {2}}}}&{\frac {2{\sqrt {3}}}{3}}&{\sqrt {2}}&2&{\sqrt {6}}+{\sqrt {2}}&\infty \\\csc &\infty &{\sqrt {6}}+{\sqrt {2}}&{\sqrt {2}}{\sqrt {2+{\sqrt {2}}}}&2&{\sqrt {2}}&{\frac {2{\sqrt {3}}}{3}}&{\sqrt {6}}-{\sqrt {2}}&1\\\end{array}}^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{12}}\\15^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{8}}\\22.5^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{6}}\\30^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{4}}\\45^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{3}}\\60^{\circ }\end{matrix}}&{\begin{matrix}{\frac {5\pi }{12}}\\75^{\circ }\end{matrix}}&{\begin{matrix}{\frac {\pi }{2}}\\90^{\circ }\end{matrix}}\\\hline \sin &0&{\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}&{\frac {\sqrt {2-{\sqrt {2}}}}{2}}&{\frac {1}{2}}&{\frac {\sqrt {2}}{2}}&{\frac {\sqrt {3}}{2}}&{\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}&1\\\cos &1&{\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}&{\frac {\sqrt {2+{\sqrt {2}}}}{2}}&{\frac {\sqrt {3}}{2}}&{\frac {\sqrt {2}}{2}}&{\frac {1}{2}}&{\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}&0\\\tan &0&2-{\sqrt {3}}&{\sqrt {2}}-1&{\frac {\sqrt {3}}{3}}&1&{\sqrt {3}}&2+{\sqrt {3}}&\infty \\\cot &\infty &2+{\sqrt {3}}&{\sqrt {2}}+1&{\sqrt {3}}&1&{\frac {\sqrt {3}}{3}}&2-{\sqrt {3}}&0\\\sec &1&{\sqrt {6}}-{\sqrt {2}}&{\sqrt {2}}{\sqrt {2-{\sqrt {2}}}}&{\frac {2{\sqrt {3}}}{3}}&{\sqrt {2}}&2&{\sqrt {6}}+{\sqrt {2}}&\infty \\\csc &\infty &{\sqrt {6}}+{\sqrt {2}}&{\sqrt {2}}{\sqrt {2+{\sqrt {2}}}}&2&{\sqrt {2}}&{\frac {2{\sqrt {3}}}{3}}&{\sqrt {6}}-{\sqrt {2}}&1\\\end{array}}}$

The symbol ∞ here represents the point at infinity on the projectively extended real line, the limit on the extended real line is +∞ on one side and -∞ on the other.

Inverse functions

Main article: Inverse trigonometric functions

The trigonometric functions are periodic, and hence not injective, so strictly they do not have an inverse function. Therefore, to define an inverse function we must restrict their domains so that the trigonometric function is bijective. In the following, the functions on the left are defined by the equation on the right; these are not proved identities. The principal inverses are usually defined as:

${\begin{array}{rrc}{\text{Function}}&{\text{Definition}}&{\text{Value Field}}\\\hline \arcsin x=y&\sin y=x&-{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}}\\\arccos x=y&\cos y=x&0\leq y\leq \pi \\\arctan x=y&\tan y=x&-{\frac {\pi }{2}}<y<{\frac {\pi }{2}}\\\operatorname {arccot} x=y&\cot y=x&0<y<\pi \\\operatorname {arcsec} x=y&\sec y=x&0\leq y\leq \pi ,\,y\neq {\frac {\pi }{2}}\\\operatorname {arccsc} x=y&\csc y=x&-{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}},\,y\neq 0\end{array}}$

The notations sin⁻¹ and cos⁻¹ are often used for arcsin and arccos, etc. When this notation is used, the inverse functions could be confused with the multiplicative inverses of the functions. The notation using the "arc-" prefix avoids such confusion, though "arcsec" for arcsecant can be confused with "arcsecond".

Just like the sine and cosine, the inverse trigonometric functions can also be defined in terms of infinite series. For example,

$\arcsin z=z+\left({\frac {1}{2}}\right){\frac {z^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{5}}{5}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {z^{7}}{7}}+\cdots \,.$

These functions may also be defined by proving that they are antiderivatives of other functions. The arcsine, for example, can be written as the following integral:

$\arcsin z=\int _{0}^{z}{\frac {1}{\sqrt {1-x^{2}}}}\,dx,\quad |z|<1.$

Analogous formulas for the other functions can be found at inverse trigonometric functions. Using the complex logarithm, one can generalize all these functions to complex arguments:

${\begin{aligned}\arcsin z&=-i\log \left(iz+{\sqrt {1-z^{2}}}\right),\,\\\arccos z&=-i\log \left(z+{\sqrt {z^{2}-1}}\right),\,\\\arctan z&={\frac {1}{2}}i\log \left({\frac {1-iz}{1+iz}}\right).\end{aligned}}$

Connection to the inner product

In an inner product space, the angle between two non-zero vectors is defined to be

$\operatorname {angle} (x,y)=\arccos {\frac {\langle x,y\rangle }{\|x\|\cdot \|y\|}}.$

Properties and applications

Main article: Uses of trigonometry

The trigonometric functions, as the name suggests, are of crucial importance in trigonometry, mainly because of the following two results.

Law of sines

The law of sines states that for an arbitrary triangle with sides a, b, and c and angles opposite those sides A, B and C:

${\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}={\frac {2\Delta }{abc}},$

where Δ is the area of the triangle, or, equivalently,

${\frac {a}{\sin A}}={\frac {b}{\sin B}}={\frac {c}{\sin C}}=2R,$

where R is the triangle's circumradius.

A Lissajous curve, a figure formed with a trigonometry-based function.

It can be proven by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in triangulation, a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.

Law of cosines

The law of cosines (also known as the cosine formula or cosine rule) is an extension of the Pythagorean theorem:

$c^{2}=a^{2}+b^{2}-2ab\cos C,\,$

or equivalently,

$\cos C={\frac {a^{2}+b^{2}-c^{2}}{2ab}}.$

In this formula the angle at C is opposite to the side c. This theorem can be proven by dividing the triangle into two right ones and using the Pythagorean theorem.

The law of cosines can be used to determine a side of a triangle if two sides and the angle between them are known. It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all the sides are known.

Law of tangents

Main article: Law of tangents

The following all form the law of tangents

${\frac {\tan {\dfrac {A-B}{2}}}{\tan {\dfrac {A+B}{2}}}}={\frac {a-b}{a+b}}\,;\qquad {\frac {\tan {\dfrac {A-C}{2}}}{\tan {\dfrac {A+C}{2}}}}={\frac {a-c}{a+c}}\,;\qquad {\frac {\tan {\dfrac {B-C}{2}}}{\tan {\dfrac {B+C}{2}}}}={\frac {b-c}{b+c}}$

The explanation of the formulae in words would be cumbersome, but the patterns of sums and differences, for the lengths and corresponding opposite angles, are apparent in the theorem.

Law of cotangents

Main article: Law of cotangents

If

$\zeta ={\sqrt {{\frac {1}{s}}(s-a)(s-b)(s-c)}}\$ (the radius of the inscribed circle for the triangle)

and

$s={\frac {a+b+c}{2}}\$ (the semi-perimeter for the triangle),

then the following all form the law of cotangents

$\cot {\frac {A}{2}}={\frac {s-a}{\zeta }}\,;\qquad \cot {\frac {B}{2}}={\frac {s-b}{\zeta }}\,;\qquad \cot {\frac {C}{2}}={\frac {s-c}{\zeta }}$

It follows that

${\frac {\cot {\dfrac {A}{2}}}{s-a}}={\frac {\cot {\dfrac {B}{2}}}{s-b}}={\frac {\cot {\dfrac {C}{2}}}{s-c}}.$

In words the theorem is: the cotangent of a half-angle equals the ratio of the semi-perimeter minus the opposite side to the said angle, to the inradius for the triangle.

Periodic functions

An animation of the additive synthesis of a square wave with an increasing number of harmonics

Sinusoidal basis functions (bottom) can form a sawtooth wave (top) when added. All the basis functions have nodes at the nodes of the sawtooth, and all but the fundamental (k = 1) have additional nodes. The oscillation seen about the sawtooth when k is large is called the Gibbs phenomenon

The trigonometric functions are also important in physics. The sine and the cosine functions, for example, are used to describe simple harmonic motion, which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. The sine and cosine functions are one-dimensional projections of uniform circular motion.

Trigonometric functions also prove to be useful in the study of general periodic functions. The characteristic wave patterns of periodic functions are useful for modeling recurring phenomena such as sound or light waves.

Under rather general conditions, a periodic function f(x) can be expressed as a sum of sine waves or cosine waves in a Fourier series. Denoting the sine or cosine basis functions by φ_k, the expansion of the periodic function f(t) takes the form:

$f(t)=\sum _{k=1}^{\infty }c_{k}\varphi _{k}(t).$

For example, the square wave can be written as the Fourier series

$f_{\text{square}}(t)={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\sin {\big (}(2k-1)t{\big )} \over 2k-1}.$

In the animation of a square wave at top right it can be seen that just a few terms already produce a fairly good approximation. The superposition of several terms in the expansion of a sawtooth wave are shown underneath.

History

Main article: History of trigonometric functions

While the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use today were developed in the medieval period. The chord function was discovered by Hipparchus of Nicaea (180–125 BCE) and Ptolemy of Roman Egypt (90–165 CE).

The functions sine and cosine can be traced to the jyā and koti-jyā functions used in Gupta period Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin.

All six trigonometric functions in current use were known in Islamic mathematics by the 9th century, as was the law of sines, used in solving triangles. al-Khwārizmī produced tables of sines, cosines and tangents.

They were studied by authors including Omar Khayyám, Bhāskara II, Nasir al-Din al-Tusi, Jamshīd al-Kāshī (14th century), Ulugh Beg (14th century), Regiomontanus (1464), Rheticus, and Rheticus' student Valentinus Otho.

Madhava of Sangamagrama (c. 1400) made early strides in the analysis of trigonometric functions in terms of infinite series.

The terms tangent and secant were first introduced by the Danish mathematician Thomas Fincke in his book Geometria rotundi (1583).

The first published use of the abbreviations sin, cos, and tan is probably by the 16th century French mathematician Albert Girard.

In a paper published in 1682, Leibniz proved that sin x is not an algebraic function of x.

Leonhard Euler's Introductio in analysin infinitorum (1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, also defining them as infinite series and presenting "Euler's formula", as well as near-modern abbreviations (sin., cos., tang., cot., sec., and cosec.).

A few functions were common historically, but are now seldom used, such as the chord (crd(θ) = 2 sin(θ/2)), the versine (versin(θ) = 1 − cos(θ) = 2 sin²(θ/2)) (which appeared in the earliest tables), the coversine (coversin(θ) = 1 − sin(θ) = versin(π/2-θ)), the haversine (haversin(θ) = 1/2versin(θ) = sin²(θ/2)), the exsecant (exsec(θ) = sec(θ) − 1), and the excosecant (excsc(θ) = exsec(π/2 − θ) = csc(θ) − 1). See List of trigonometric identities for more relations between these functions.

Etymology

The word sine derives from Latin sinus, meaning "bend; bay", and more specifically "the hanging fold of the upper part of a toga", "the bosom of a garment", which was chosen as the translation of what was interpreted as the Arabic word jaib, meaning "pocket" or "fold" in the twelfth-century translations of works by Al-Battani and al-Khwārizmī into Medieval Latin. The choice was based on a misreading of the Arabic written form j-y-b (جيب), which itself originated as a transliteration from Sanskrit jīvā, which along with its synonym jyā (the standard Sanskrit term for the sine) translates to "bowstring", being in turn adopted from Ancient Greek χορδή "string".

The word tangent comes from Latin tangens meaning "touching", since the line touches the circle of unit radius, whereas secant stems from Latin secans—"cutting"—since the line cuts the circle.

The prefix "co-" (in "cosine", "cotangent", "cosecant") is found in Edmund Gunter's Canon triangulorum (1620), which defines the cosinus as an abbreviation for the sinus complementi (sine of the complementary angle) and proceeds to define the cotangens similarly.

See also

• All Students Take Calculus — a mnemonic for recalling the signs of trigonometric functions in a particular quadrant of a Cartesian plane
• Aryabhata's sine table
• Bhaskara I's sine approximation formula
• Generalized trigonometry
• Generating trigonometric tables
• Hyperbolic function
• List of periodic functions
• List of trigonometric identities
• Madhava series
• Madhava's sine table
• Polar sine — a generalization to vertex angles
• Proofs of trigonometric identities
• Versine — for several less used trigonometric functions

References

• Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253. 
• Lars Ahlfors, Complex Analysis: an introduction to the theory of analytic functions of one complex variable, second edition, McGraw-Hill Book Company, New York, 1966.
• Boyer, Carl B., A History of Mathematics, John Wiley & Sons, Inc., 2nd edition. (1991). ISBN 0-471-54397-7.
• Gal, Shmuel and Bachelis, Boris. An accurate elementary mathematical library for the IEEE floating point standard, ACM Transactions on Mathematical Software (1991).
• Joseph, George G., The Crest of the Peacock: Non-European Roots of Mathematics, 2nd ed. Penguin Books, London. (2000). ISBN 0-691-00659-8.
• Kantabutra, Vitit, "On hardware for computing exponential and trigonometric functions," IEEE Trans. Computers 45 (3), 328–339 (1996).
• Maor, Eli, Trigonometric Delights, Princeton Univ. Press. (1998). Reprint edition (February 25, 2002): ISBN 0-691-09541-8.
• Needham, Tristan, "Preface"" to Visual Complex Analysis. Oxford University Press, (1999). ISBN 0-19-853446-9.
• Nielsen, Kaj L. (1966), Logarithmic and Trigonometric Tables to Five Places (2nd ed.), New York, USA: Barnes & Noble, LCCN 61-9103 
• O'Connor, J. J., and E. F. Robertson, "Trigonometric functions", MacTutor History of Mathematics archive. (1996).
• O'Connor, J. J., and E. F. Robertson, "Madhava of Sangamagramma", MacTutor History of Mathematics archive. (2000).
• Pearce, Ian G., "Madhava of Sangamagramma", MacTutor History of Mathematics archive. (2002).
• Weisstein, Eric W., "Tangent" from MathWorld, accessed 21 January 2006.

External links

• Hazewinkel, Michiel, ed. (2001) [1994], "Trigonometric functions", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 
• Visionlearning Module on Wave Mathematics
• GonioLab Visualization of the unit circle, trigonometric and hyperbolic functions
• q-Sine Article about the q-analog of sin at MathWorld
• q-Cosine Article about the q-analog of cos at MathWorld

English Journal

• Analytic representations of bath correlation functions for ohmic and superohmic spectral densities using simple poles.

• Ritschel G1, Eisfeld A1.
• The Journal of chemical physics.J Chem Phys.2014 Sep 7;141(9):094101. doi: 10.1063/1.4893931.
• We present a scheme to express a bath correlation function (BCF) corresponding to a given spectral density (SD) as a sum of damped harmonic oscillations. Such a representation is needed, for example, in many open quantum system approaches. To this end we introduce a class of fit functions that enabl
• PMID 25194358

• Novel tablet formulation of amorphous candesartan cilexetil solid dispersions involving P-gp inhibition for optimal drug delivery: in vitro and in vivo evaluation.

• Surampalli G1, Nanjwade BK, Patil PA, Chilla R.
• Drug delivery.Drug Deliv.2014 Jul 31:1-15. [Epub ahead of print]
• Abstract Objective: The aim of this study was to develop a novel tablet formulation of amorphous candesartan cilexetil (CAN) solid dispersion involving effective P-gp inhibition for optimal drug delivery by direct compression (DC) method. Methods: To accomplish DC, formulation blends were evaluated
• PMID 25080228

• Longitudinal and transverse Hall resistivities in NaFe₁-xCoxAs single crystals with x = 0.022 and 0.0205: weak pinning and anomalous electrical transport properties.

• Wang LM1, Wang CY, Sou UC, Yang HC, Chang LJ, Redding C, Song Y, Dai P, Zhang C.
• Journal of physics. Condensed matter : an Institute of Physics journal.J Phys Condens Matter.2013 Oct 2;25(39):395702. doi: 10.1088/0953-8984/25/39/395702. Epub 2013 Sep 4.
• The in-plane longitudinal and Hall resistivities, ρxx and ρxy, of superconducting NaFe1-xCoxAs (NFCA) single crystals with x = 0.022 and 0.0205 in the mixed state and the normal state were measured to study the electrical transport properties in nearly optimum-doping iron-based superconductors. Th
• PMID 24002242

Japanese Journal

• 22aDC-7 超対称性を用いた例外群エルミート対称空間の余接束の構築

• 日本物理学会講演概要集 70(1), 15, 2015-03-21
• NAID 110009990415

• Characteristic Cycle and the Euler Number of a Constructible Sheaf on a Surface

• Journal of Mathematical Sciences, The University of Tokyo 22(1), 2015-02-27
• NAID 120005713009

• Symplectic homology of disc cotangent bundles of domains in Euclidean space

• Journal of Symplectic Geometry 12(3), 511-552, 2014-09
• NAID 120005469578

Related Links

• ぽあ (cotangent) on Twitter

The latest from ぽあ (@cotangent). 青色を推すことが多いです。 黒ネコ同盟/アイカツ!/石原夏織/瀬戸麻沙美/花澤香菜/JAM Project/tiaraway/木崎湖/小諸/水曜どうでしょう/ロード乗り/. ドリームアカデミー

• Cotangent -- from Wolfram MathWorld

The cotangent function cotz is the function defined by cotz = 1/(tanz) (1) = (i(e^(iz)+e^(-iz)))/(e^(iz)-e^(-iz)) (2) = (i(e^(2iz)+1))/(e^(2iz)-1), (3) where tanz is the tangent. The cotangent is implemented in Mathematica as Cot[z]. The ...

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