sine function

正弦関数

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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2019/05/26 00:12:04」(JST)

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For other uses, see Sine (disambiguation).

Not to be confused with sign, sign (mathematics), or sin.

Sine
Basic features
Parity	odd
Domain	(−∞, +∞) a
Codomain	[−1, 1] a
Period	2π
Specific values
At zero	0
Maxima	(2kπ + π/2, 1)b
Minima	(2kπ − π/2, −1)
Specific features
Root	kπ
Critical point	kπ + π/2
Inflection point	kπ
Fixed point	0
a For real numbers. b Variable k is an integer.

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

In mathematics, the sine is a trigonometric function of an angle. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse).

More generally, the definition of sine (and other trigonometric functions) can be extended to any real value in terms of the length of a certain line segment in a unit circle. More modern definitions express the sine as an infinite series or as the solution of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers.

The sine function is commonly used to model periodic phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year.

The function sine can be traced to the jyā and koṭi-jyā functions used in Gupta period Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin. The word "sine" (Latin "sinus") comes from a Latin mistranslation of the Arabic jiba, which is a transliteration of the Sanskrit word for half the chord, jya-ardha.

Right-angled triangle definition

For the angle α, the sine function gives the ratio of the length of the opposite side to the length of the hypotenuse.

To define the sine function of an acute angle α, start with a right triangle that contains an angle of measure α; in the accompanying figure, angle A in triangle ABC is the angle of interest. The three sides of the triangle are named as follows:

• The opposite side is the side opposite to the angle of interest, in this case side a.
• 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 adjacent side is the remaining side, in this case side b. It forms a side of (is adjacent to) both the angle of interest (angle A) and the right angle.

Once such a triangle is chosen, the sine of the angle is equal to the length of the opposite side divided by the length of the hypotenuse, or:

$\sin(\alpha )={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}$

The other trigonometric functions of the angle can be defined similarly; for example, the cosine of the angle is the ratio between the adjacent side and the hypotenuse, while the tangent gives the ratio between the opposite and adjacent sides.

As stated, the value $$

 
   
     
       sin
       ⁡
       (
       α
       )
     
   
   {\displaystyle \sin(\alpha )}
 

appears to depend on the choice of right triangle containing an angle of measure α. However, this is not the case: all such triangles are similar, and so the ratio is the same for each of them.

Unit circle definition

Unit circle: the radius has length 1. The variable t measures the angle referred to as θ in the text.

In trigonometry, a unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinate system.

Let a line through the origin, making an angle of θ with the positive half of the x-axis, intersect the unit circle. The x- and y-coordinates of this point of intersection are equal to cos(θ) and sin(θ), respectively. The point's distance from the origin is always 1.

Unlike the definitions with the right triangle or slope, the angle can be extended to the full set of real arguments by using the unit circle. This can also be achieved by requiring certain symmetries and that sine be a periodic function.

Animation showing how the sine function (in red) $$

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

is graphed from the y-coordinate (red dot) of a point on the unit circle (in green) at an angle of θ.

Identities

Main article: List of trigonometric identities

Exact identities (using radians):

These apply for all values of $$

 
   
     
       θ
     
   
   {\displaystyle \theta }
 

.

$\sin(\theta )=\cos \left({\frac {\pi }{2}}-\theta \right)={\frac {1}{\csc(\theta )}}$

Reciprocal

The reciprocal of sine is cosecant, i.e., the reciprocal of sin(A) is csc(A), or cosec(A). Cosecant gives the ratio of the length of the hypotenuse to the length of the opposite side:

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

Inverse

The usual principal values of the arcsin(x) function graphed on the cartesian plane. Arcsin is the inverse of sin.

The inverse function of sine is arcsine (arcsin or asin) or inverse sine (sin^-1). As sine is non-injective, it is not an exact inverse function but a partial inverse function. For example, sin(0) = 0, but also sin(π) = 0, sin(2π) = 0 etc. It follows that the arcsine function is multivalued: arcsin(0) = 0, but also arcsin(0) = π, arcsin(0) = 2π, etc. When only one value is desired, the function may be restricted to its principal branch. With this restriction, for each x in the domain the expression arcsin(x) will evaluate only to a single value, called its principal value.

$\theta =\arcsin \left({\frac {\text{opposite}}{\text{hypotenuse}}}\right)=\sin ^{-1}\left({\frac {a}{h}}\right).$

k is some integer:

${\begin{aligned}\sin(y)=x\iff &y=\arcsin(x)+2\pi k,{\text{ or }}\\&y=\pi -\arcsin(x)+2\pi k\end{aligned}}$

Or in one equation:

$\sin(y)=x\iff y=(-1)^{k}\arcsin(x)+\pi k$

Arcsin satisfies:

$\sin(\arcsin(x))=x\!$

and

$\arcsin(\sin(\theta ))=\theta \quad {\text{for }}-{\frac {\pi }{2}}\leq \theta \leq {\frac {\pi }{2}}.$

Calculus

See also: List of integrals of trigonometric functions and Differentiation of trigonometric functions

For the sine function:

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

The derivative is:

$f'(x)=\cos(x)$

The antiderivative is:

$\int f(x)\,dx=-\cos(x)+C$

C denotes the constant of integration.

Other trigonometric functions

The sine and cosine functions are related in multiple ways. The two functions are out of phase by 90°: $$

 
   
     
       sin
       ⁡
       (
       π
       
         /
       
       2
       −
       x
       )
     
   
   {\displaystyle \sin(\pi /2-x)}
 

= $$

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

for all angles x. Also, the derivative of the function sin(x) is cos(x).

It is possible to express any trigonometric function in terms of any other (up to a plus or minus sign, or using the sign function).

Sine in terms of the other common trigonometric functions:

	f θ	Using plus/minus (±)					Using sign function (sgn)
		f θ =	± per Quadrant				f θ =
			I	II	III	IV
cos	$$ sin ⁡ ( θ ) {\displaystyle \sin(\theta )}	$$ = ± 1 − cos 2 ⁡ ( θ ) {\displaystyle =\pm {\sqrt {1-\cos ^{2}(\theta )}}}	+	+	−	−	$$ = sgn ⁡ ( cos ⁡ ( θ − π 2 ) ) 1 − cos 2 ⁡ ( θ ) {\displaystyle =\operatorname {sgn} \left(\cos \left(\theta -{\frac {\pi }{2}}\right)\right){\sqrt {1-\cos ^{2}(\theta )}}}
	$$ cos ⁡ ( θ ) {\displaystyle \cos(\theta )}	$$ = ± 1 − sin 2 ⁡ ( θ ) {\displaystyle =\pm {\sqrt {1-\sin ^{2}(\theta )}}}	+	−	−	+	$$ = sgn ⁡ ( sin ⁡ ( θ + π 2 ) ) 1 − sin 2 ⁡ ( θ ) {\displaystyle =\operatorname {sgn} \left(\sin \left(\theta +{\frac {\pi }{2}}\right)\right){\sqrt {1-\sin ^{2}(\theta )}}}
cot	$$ sin ⁡ ( θ ) {\displaystyle \sin(\theta )}	$$ = ± 1 1 + cot 2 ⁡ ( θ ) {\displaystyle =\pm {\frac {1}{\sqrt {1+\cot ^{2}(\theta )}}}}	+	+	−	−	$$ = sgn ⁡ ( cot ⁡ ( θ 2 ) ) 1 1 + cot 2 ⁡ ( θ ) {\displaystyle =\operatorname {sgn} \left(\cot \left({\frac {\theta }{2}}\right)\right){\frac {1}{\sqrt {1+\cot ^{2}(\theta )}}}}
	$$ cot ⁡ ( θ ) {\displaystyle \cot(\theta )}	$$ = ± 1 − sin 2 ⁡ ( θ ) sin ⁡ ( θ ) {\displaystyle =\pm {\frac {\sqrt {1-\sin ^{2}(\theta )}}{\sin(\theta )}}}	+	−	−	+	$$ = sgn ⁡ ( sin ⁡ ( θ + π 2 ) ) 1 − sin 2 ⁡ ( θ ) sin ⁡ ( θ ) {\displaystyle =\operatorname {sgn} \left(\sin \left(\theta +{\frac {\pi }{2}}\right)\right){\frac {\sqrt {1-\sin ^{2}(\theta )}}{\sin(\theta )}}}
tan	$$ sin ⁡ ( θ ) {\displaystyle \sin(\theta )}	$$ = ± tan ⁡ ( θ ) 1 + tan 2 ⁡ ( θ ) {\displaystyle =\pm {\frac {\tan(\theta )}{\sqrt {1+\tan ^{2}(\theta )}}}}	+	−	−	+	$$ = sgn ⁡ ( tan ⁡ ( 2 θ + π 4 ) ) tan ⁡ ( θ ) 1 + tan 2 ⁡ ( θ ) {\displaystyle =\operatorname {sgn} \left(\tan \left({\frac {2\theta +\pi }{4}}\right)\right){\frac {\tan(\theta )}{\sqrt {1+\tan ^{2}(\theta )}}}}
	$$ tan ⁡ ( θ ) {\displaystyle \tan(\theta )}	$$ = ± sin ⁡ ( θ ) 1 − sin 2 ⁡ ( θ ) {\displaystyle =\pm {\frac {\sin(\theta )}{\sqrt {1-\sin ^{2}(\theta )}}}}	+	−	−	+	$$ = sgn ⁡ ( sin ⁡ ( θ + π 2 ) ) sin ⁡ ( θ ) 1 − sin 2 ⁡ ( θ ) {\displaystyle =\operatorname {sgn} \left(\sin \left(\theta +{\frac {\pi }{2}}\right)\right){\frac {\sin(\theta )}{\sqrt {1-\sin ^{2}(\theta )}}}}
sec	$$ sin ⁡ ( θ ) {\displaystyle \sin(\theta )}	$$ = ± sec 2 ⁡ ( θ ) − 1 sec ⁡ ( θ ) {\displaystyle =\pm {\frac {\sqrt {\sec ^{2}(\theta )-1}}{\sec(\theta )}}}	+	−	+	−	$$ = sgn ⁡ ( sec ⁡ ( 4 θ − π 2 ) ) sec 2 ⁡ ( θ ) − 1 sec ⁡ ( θ ) {\displaystyle =\operatorname {sgn} \left(\sec \left({\frac {4\theta -\pi }{2}}\right)\right){\frac {\sqrt {\sec ^{2}(\theta )-1}}{\sec(\theta )}}}
	$$ sec ⁡ ( θ ) {\displaystyle \sec(\theta )}	$$ = ± 1 1 − sin 2 ⁡ ( θ ) {\displaystyle =\pm {\frac {1}{\sqrt {1-\sin ^{2}(\theta )}}}}	+	−	−	+	$$ = sgn ⁡ ( sin ⁡ ( θ + π 2 ) ) 1 1 − sin 2 ⁡ ( θ ) {\displaystyle =\operatorname {sgn} \left(\sin \left(\theta +{\frac {\pi }{2}}\right)\right){\frac {1}{\sqrt {1-\sin ^{2}(\theta )}}}}

Note that for all equations which use plus/minus (±), the result is positive for angles in the first quadrant.

The basic relationship between the sine and the cosine can also be expressed as the Pythagorean trigonometric identity:

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

where sin²(x) means (sin(x))².

Properties relating to the quadrants

The four quadrants of a Cartesian coordinate system.

The table below displays many of the key properties of the sine function (sign, monotonicity, convexity), arranged by the quadrant of the argument. For arguments outside those in the table, one may compute the corresponding information by using the periodicity $\sin(\alpha +360^{\circ })=\sin(\alpha )$ of the sine function.

Quadrant	Degrees	Radians	Value	Sign	Monotony	Convexity
1st Quadrant	0 ∘ < x < 90 ∘ {\displaystyle 0^{\circ }<x<90^{\circ }}	0 < x < π 2 {\displaystyle 0<x<{\frac {\pi }{2}}}	0 < sin ⁡ ( x ) < 1 {\displaystyle 0<\sin(x)<1}	$$ + {\displaystyle +}	increasing	concave
2nd Quadrant	90 ∘ < x < 180 ∘ {\displaystyle 90^{\circ }<x<180^{\circ }}	π 2 < x < π {\displaystyle {\frac {\pi }{2}}<x<\pi }	0 < sin ⁡ ( x ) < 1 {\displaystyle 0<\sin(x)<1}	$$ + {\displaystyle +}	decreasing	concave
3rd Quadrant	180 ∘ < x < 270 ∘ {\displaystyle 180^{\circ }<x<270^{\circ }}	π < x < 3 π 2 {\displaystyle \pi <x<{\frac {3\pi }{2}}}	− 1 < sin ⁡ ( x ) < 0 {\displaystyle -1<\sin(x)<0}	$$ − {\displaystyle -}	decreasing	convex
4th Quadrant	270 ∘ < x < 360 ∘ {\displaystyle 270^{\circ }<x<360^{\circ }}	3 π 2 < x < 2 π {\displaystyle {\frac {3\pi }{2}}<x<2\pi }	− 1 < sin ⁡ ( x ) < 0 {\displaystyle -1<\sin(x)<0}	$$ − {\displaystyle -}	increasing	convex

The quadrants of the unit circle and of sin(x), using the Cartesian coordinate system.

The following table gives basic information at the boundary of the quadrants.

Degrees	Radians	$$ sin ⁡ ( x ) {\displaystyle \sin(x)}	Point type
$$ 0 ∘ {\displaystyle 0^{\circ }}	$$ 0 {\displaystyle 0}	$$ 0 {\displaystyle 0}	Root, Inflection
$$ 90 ∘ {\displaystyle 90^{\circ }}	$$ π 2 {\displaystyle {\frac {\pi }{2}}}	$$ 1 {\displaystyle 1}	Maximum
$$ 180 ∘ {\displaystyle 180^{\circ }}	$$ π {\displaystyle \pi }	$$ 0 {\displaystyle 0}	Root, Inflection
$$ 270 ∘ {\displaystyle 270^{\circ }}	$$ 3 π 2 {\displaystyle {\frac {3\pi }{2}}}	$$ − 1 {\displaystyle -1}	Minimum

Series definition

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

This animation shows how including more and more terms in the partial sum of its Taylor series approaches a sine curve.

Using only geometry and properties of limits, it can be shown that the derivative of sine is cosine, and that the derivative of cosine is the negative of sine.

Using the reflection from the calculated geometric derivation of the sine is with the (4n+k)-th derivative at the point 0:

$\sin ^{(4n+k)}(0)={\begin{cases}0&{\text{when }}k=0\\1&{\text{when }}k=1\\0&{\text{when }}k=2\\-1&{\text{when }}k=3\end{cases}}$

This gives the following Taylor series expansion at x = 0. One can then use the theory of Taylor series to show that the following identities hold for all real numbers x (where x is the angle in radians) :

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

If x were expressed in degrees then the series would contain factors involving powers of π/180: if x is the number of degrees, the number of radians is y = πx /180, so

${\begin{aligned}\sin(x_{\mathrm {deg} })&=\sin(y_{\mathrm {rad} })\\&={\frac {\pi }{180}}x-\left({\frac {\pi }{180}}\right)^{3}{\frac {x^{3}}{3!}}+\left({\frac {\pi }{180}}\right)^{5}{\frac {x^{5}}{5!}}-\left({\frac {\pi }{180}}\right)^{7}{\frac {x^{7}}{7!}}+\cdots .\end{aligned}}$

The series formulas for the sine and cosine are uniquely determined, up to the choice of unit for angles, by the requirements that

${\begin{aligned}\sin(0)=0&{\text{ and }}\sin(2x)=2\sin(x)\cos(x)\\\cos ^{2}(x)+\sin ^{2}(x)=1&{\text{ and }}\cos(2x)=\cos ^{2}(x)-\sin ^{2}(x)\\\end{aligned}}$

The radian is the unit that leads to the expansion with leading coefficient 1 for the sine and is determined by the additional requirement that

$\sin(x)\approx x{\text{ when }}x\approx 0.$

The coefficients for both the sine and cosine series may therefore be derived by substituting their expansions into the pythagorean and double angle identities, taking the leading coefficient for the sine to be 1, and matching the remaining coefficients.

In general, mathematically important relationships between the sine and cosine functions and the exponential function (see, for example, Euler's formula) are substantially simplified when angles are expressed in radians, rather than in degrees, grads or other units. Therefore, in most branches of mathematics beyond practical geometry, angles are generally assumed to be expressed in radians.

A similar series is Gregory's series for arctan, which is obtained by omitting the factorials in the denominator.

Continued fraction

The sine function can also be represented as a generalized continued fraction:

$\sin(x)={\cfrac {x}{1+{\cfrac {x^{2}}{2\cdot 3-x^{2}+{\cfrac {2\cdot 3x^{2}}{4\cdot 5-x^{2}+{\cfrac {4\cdot 5x^{2}}{6\cdot 7-x^{2}+\ddots }}}}}}}}.$

The continued fraction representation can be derived from Euler's continued fraction formula and expresses the real number values, both rational and irrational, of the sine function.

Fixed point

The fixed point iteration x_n+1 = sin(x_n) with initial value x₀ = 2 converges to 0.

Zero is the only real fixed point of the sine function; in other words the only intersection of the sine function and the identity function is sin(0) = 0.

Arc length

The arc length of the sine curve between $a$ and $b$ is $\int _{a}^{b}\!{\sqrt {1+\cos ^{2}(x)}}\,dx$. This integral is an elliptic integral of the second kind.

The arc length for a full period is $$

 
   
     
       
         
           
             4
             
               
                 2
                 
                   π
                   
                     3
                   
                 
               
             
           
           
             Γ
             (
             1
             
               /
             
             4
             
               )
               
                 2
               
             
           
         
       
       +
       
         
           
             Γ
             (
             1
             
               /
             
             4
             
               )
               
                 2
               
             
           
           
             2
             π
           
         
       
       =
       7.640395578
       …
     
   
   {\displaystyle {\frac {4{\sqrt {2\pi ^{3}}}}{\Gamma (1/4)^{2}}}+{\frac {\Gamma (1/4)^{2}}{\sqrt {2\pi }}}=7.640395578\ldots }
 

where $$

 
   
     
       Γ
     
   
   {\displaystyle \Gamma }
 

is the Gamma function.

The arc length of the sine curve from 0 to x is the above number divided by $$

 
   
     
       2
       π
     
   
   {\displaystyle 2\pi }
 

times x, plus a correction that varies periodically in x with period $$

 
   
     
       π
     
   
   {\displaystyle \pi }
 

. The Fourier series for this correction can be written in closed form using special functions, but it is perhaps more instructive to write the decimal approximations of the Fourier coefficients. The sine curve arc length from 0 to x is

$1.21600672x+0.10317093\sin(2x)-0.00220445\sin(4x)+0.00012584\sin(6x)-0.00001011\sin(8x)+\cdots$

Law of sines

Main article: 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:

${\frac {\sin A}{a}}={\frac {\sin B}{b}}={\frac {\sin C}{c}}.$

This is equivalent to the equality of the first three expressions below:

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

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.

Special values

See also: Exact trigonometric constants

Some common angles (θ) shown on the unit circle. The angles are given in degrees and radians, together with the corresponding intersection point on the unit circle, (cos(θ), sin(θ)).

For certain integral numbers x of degrees, the value of sin(x) is particularly simple. A table of some of these values is given below.

x (angle)				sin(x)
Degrees	Radians	Gradians	Turns	Exact	Decimal
0°	0	0g	0	0	0
180°	π	200g	1/2
15°	1/12π	16 2/3g	1/24	$$ 6 − 2 4 {\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}	0.258819045102521
165°	11/12π	183 1/3g	11/24
30°	1/6π	33 1/3g	1/12	1/2	0.5
150°	5/6π	166 2/3g	5/12
45°	1/4π	50g	1/8	$$ 2 2 {\displaystyle {\frac {\sqrt {2}}{2}}}	0.707106781186548
135°	3/4π	150g	3/8
60°	1/3π	66 2/3g	1/6	$$ 3 2 {\displaystyle {\frac {\sqrt {3}}{2}}}	0.866025403784439
120°	2/3π	133 1/3g	1/3
75°	5/12π	83 1/3g	5/24	$$ 6 + 2 4 {\displaystyle {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}	0.965925826289068
105°	7/12π	116 2/3g	7/24
90°	1/2π	100g	1/4	1	1

90 degree increments:

x in degrees	0°	90°	180°	270°	360°
x in radians	0	π/2	π	3π/2	2π
x in gons	0	100g	200g	300g	400g
x in turns	0	1/4	1/2	3/4	1
sin x	0	1	0	-1	0

Other values not listed above:

$\sin \left({\frac {\pi }{60}}\right)=\sin(3^{\circ })={\frac {(2-{\sqrt {12}}){\sqrt {5+{\sqrt {5}}}}+({\sqrt {10}}-{\sqrt {2}})({\sqrt {3}}+1)}{16}}$ OEIS: A019812

$\sin \left({\frac {\pi }{30}}\right)=\sin(6^{\circ })={\frac {{\sqrt {30-{\sqrt {180}}}}-{\sqrt {5}}-1}{8}}$ OEIS: A019815

$\sin \left({\frac {\pi }{20}}\right)=\sin(9^{\circ })={\frac {{\sqrt {10}}+{\sqrt {2}}-{\sqrt {20-{\sqrt {80}}}}}{8}}$ OEIS: A019818

$\sin \left({\frac {\pi }{15}}\right)=\sin(12^{\circ })={\frac {{\sqrt {10+{\sqrt {20}}}}+{\sqrt {3}}-{\sqrt {15}}}{8}}$ OEIS: A019821

$\sin \left({\frac {\pi }{10}}\right)=\sin(18^{\circ })={\frac {{\sqrt {5}}-1}{4}}={\tfrac {1}{2}}\varphi ^{-1}$ OEIS: A019827

$\sin \left({\frac {7\pi }{60}}\right)=\sin(21^{\circ })={\frac {(2+{\sqrt {12}}){\sqrt {5-{\sqrt {5}}}}-({\sqrt {10}}+{\sqrt {2}})({\sqrt {3}}-1)}{16}}$ OEIS: A019830

$\sin \left({\frac {\pi }{8}}\right)=\sin(22.5^{\circ })={\frac {\sqrt {2-{\sqrt {2}}}}{2}}$

$\sin \left({\frac {2\pi }{15}}\right)=\sin(24^{\circ })={\frac {{\sqrt {3}}+{\sqrt {15}}-{\sqrt {10-{\sqrt {20}}}}}{8}}$ OEIS: A019833

$\sin \left({\frac {3\pi }{20}}\right)=\sin(27^{\circ })={\frac {{\sqrt {20+{\sqrt {80}}}}-{\sqrt {10}}+{\sqrt {2}}}{8}}$ OEIS: A019836

$\sin \left({\frac {11\pi }{60}}\right)=\sin(33^{\circ })={\frac {({\sqrt {12}}-2){\sqrt {5+{\sqrt {5}}}}+({\sqrt {10}}-{\sqrt {2}})({\sqrt {3}}+1)}{16}}$ OEIS: A019842

$\sin \left({\frac {\pi }{5}}\right)=\sin(36^{\circ })={\frac {\sqrt {10-{\sqrt {20}}}}{4}}$ OEIS: A019845

$\sin \left({\frac {13\pi }{60}}\right)=\sin(39^{\circ })={\frac {(2-{\sqrt {12}}){\sqrt {5-{\sqrt {5}}}}+({\sqrt {10}}+{\sqrt {2}})({\sqrt {3}}+1)}{16}}$ OEIS: A019848

$\sin \left({\frac {7\pi }{30}}\right)=\sin(42^{\circ })={\frac {{\sqrt {30+{\sqrt {180}}}}-{\sqrt {5}}+1}{8}}$ OEIS: A019851

Relationship to complex numbers

Main article: Trigonometric functions § Relationship to exponential function and complex numbers

An illustration of the complex plane. The imaginary numbers are on the vertical coordinate axis.

Sine is used to determine the imaginary part of a complex number given in polar coordinates (r, φ):

$z=r(\cos(\varphi )+i\sin(\varphi ))$

the imaginary part is:

$\operatorname {Im} (z)=r\sin(\varphi )$

r and φ represent the magnitude and angle of the complex number respectively. i is the imaginary unit. z is a complex number.

Although dealing with complex numbers, sine's parameter in this usage is still a real number. Sine can also take a complex number as an argument.

Sine with a complex argument

$$

 
   
     
       sin
       ⁡
       (
       z
       )
     
   
   {\displaystyle \sin(z)}
 

Domain coloring of sin(z) in the complex plane. Brightness indicates absolute magnitude, saturation represents complex argument.

sin(z) as a vector field

$$

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

is the imaginary part of $$

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

.

The definition of the sine function for complex arguments z:

${\begin{aligned}\sin(z)&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}z^{2n+1}\\&={\frac {e^{iz}-e^{-iz}}{2i}}\\&={\frac {\sinh \left(iz\right)}{i}}\end{aligned}}$

where i ² = −1, and sinh is hyperbolic sine. This is an entire function. Also, for purely real x,

$\sin(x)=\operatorname {Im} (e^{ix}).$

For purely imaginary numbers:

$\sin(iy)=i\sinh(y).$

It is also sometimes useful to express the complex sine function in terms of the real and imaginary parts of its argument:

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

Partial fraction and product expansions of complex sine

Using the partial fraction expansion technique in complex analysis, one can find that the infinite series

${\begin{aligned}\sum _{n=-\infty }^{\infty }{\frac {(-1)^{n}}{z-n}}={\frac {1}{z}}-2z\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n^{2}-z^{2}}}\end{aligned}}$

both converge and are equal to ${\frac {\pi }{\sin \pi z}}$. Similarly, one can show that

${\begin{aligned}{\frac {\pi ^{2}}{\sin ^{2}(\pi z)}}=\sum _{n=-\infty }^{\infty }{\frac {1}{(z-n)^{2}}}.\end{aligned}}$

Using product expansion technique, one can derive

${\begin{aligned}\sin(\pi z)=\pi z\prod _{n=1}^{\infty }{\Bigl (}1-{\frac {z^{2}}{n^{2}}}{\Bigr )}.\end{aligned}}$

Usage of complex sine

sin(z) is found in the functional equation for the Gamma function,

$\Gamma (s)\Gamma (1-s)={\pi \over \sin(\pi s)},$

which in turn is found in the functional equation for the Riemann zeta-function,

$\zeta (s)=2(2\pi )^{s-1}\Gamma (1-s)\sin(\pi s/2)\zeta (1-s).$

As a holomorphic function, sin z is a 2D solution of Laplace's equation:

$\Delta u(x_{1},x_{2})=0.$

It is also related with level curves of pendulum.

Complex graphs

Sine function in the complex plane

real component	imaginary component	magnitude

Arcsine function in the complex plane

real component	imaginary component	magnitude

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 function sine (and cosine) can be traced to the jyā and koṭi-jyā functions used in Gupta period (320 to 550 CE) Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin.

The first published use of the abbreviations 'sin', 'cos', and 'tan' is by the 16th century French mathematician Albert Girard; these were further promulgated by Euler (see below). The Opus palatinum de triangulis of Georg Joachim Rheticus, a student of Copernicus, was probably the first in Europe to define trigonometric functions directly in terms of right triangles instead of circles, with tables for all six trigonometric functions; this work was finished by Rheticus' student Valentin Otho in 1596.

In a paper published in 1682, Leibniz proved that sin x is not an algebraic function of x. Roger Cotes computed the derivative of sine in his Harmonia Mensurarum (1722). 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 the near-modern abbreviations sin., cos., tang., cot., sec., and cosec.

Etymology

Etymologically, the word sine derives from the Sanskrit word for chord, jiva*(jya being its more popular synonym). This was transliterated in Arabic as jiba جيب, which however is meaningless in that language and abbreviated jb جب . Since Arabic is written without short vowels, "jb" was interpreted as the word jaib جيب, which means "bosom". When the Arabic texts were translated in the 12th century into Latin by Gerard of Cremona, he used the Latin equivalent for "bosom", sinus (which means "bosom" or "bay" or "fold"). Gerard was probably not the first scholar to use this translation; Robert of Chester appears to have preceded him and there is evidence of even earlier usage. The English form sine was introduced in the 1590s.

Software implementations

See also: Lookup table § Computing sines

The sine function, along with other trigonometric functions, is widely available across programming languages and platforms. In computing, it is typically abbreviated to sin.

Some CPU architectures have a built-in instruction for sine, including the Intel x87 FPUs since the 80387.

In programming languages, sin is typically either a built-in function or found within the language's standard math library.

For example, the C standard library defines sine functions within math.h: sin(double), sinf(float), and sinl(long double). The parameter of each is a floating point value, specifying the angle in radians. Each function returns the same data type as it accepts. Many other trigonometric functions are also defined in math.h, such as for cosine, arc sine, and hyperbolic sine (sinh).

Similarly, Python defines math.sin(x) within the built-in math module. Complex sine functions are also available within the cmath module, e.g. cmath.sin(z). CPython's math functions call the C math library, and use a double-precision floating-point format.

There is no standard algorithm for calculating sine. IEEE 754-2008, the most widely used standard for floating-point computation, does not address calculating trigonometric functions such as sine. Algorithms for calculating sine may be balanced for such constraints as speed, accuracy, portability, or range of input values accepted. This can lead to different results for different algorithms, especially for special circumstances such as very large inputs, e.g. sin(1022).

A once common programming optimization, used especially in 3D graphics, was to pre-calculate a table of sine values, for example one value per degree. This allowed results to be looked up from a table rather than being calculated in real time. With modern CPU architectures this method may offer no advantage.

The CORDIC algorithm is commonly used in scientific calculators.

Turns based implementations

Some software libraries provide implementations of sine using the input angle in half-Turns. Representing angles in Turns or half-Turns has accuracy advantages and efficiency advantages in some cases.

Environment	function name	angle units
MATLAB	sinpi	half-Turns
OpenCL	sinpi	half-Turns
R	sinpi	half-Turns
Julia	sinpi	half-Turns
CUDA	sinpi	half-Turns
ARM	sinpi	half-Turns

The accuracy advantage stems from the ability to perfectly represent key angles like full-Turn, half-Turn, and quarter-Turn losslessly in binary floating-point or fixed-point. In contrast, representing 2*pi, pi, and pi/2 in binary floating-point or binary scaled fixed-point always involves a loss of accuracy.

Turns also have an accuracy advantage and efficiency advantage for computing modulo to one period. Computing modulo 1 Turn or modulo 2 half-Turns can be losslessly and efficiently computed in both floating-point and fixed-point. For example, computing modulo 1 or modulo 2 for a binary point scaled fixed-point value requires only a bit shift or bitwise AND operation. In contrast, computing modulo 2*pi involves inaccuracies in representing 2*pi.

For applications involving angle sensors, the sensor typically provides angle measurements in a form directly compatible with Turns or half-Turns

For example, an angle sensor may count from 0 to 4096 over one complete revolution. If half-Turns are used as the unit for angle, then the value provided by the sensor directly and losslessly maps to a fixed-point data type with 11 bits to the right of the binary point. In contrast, if Radians are used as the unit for storing the angle, then the inaccuracies and cost of multiplying the raw sensor integer by an approximation to pi/2048 would be incurred.

See also

• Āryabhaṭa's sine table
• Bhaskara I's sine approximation formula
• Discrete sine transform
• Euler's formula
• Generalized trigonometry
• Hyperbolic function
• Law of sines
• List of periodic functions
• List of trigonometric identities
• Madhava series
• Madhava's sine table
• Optical sine theorem
• Polar sine — a generalization to vertex angles
• Proofs of trigonometric identities
• Sinc function
• Sine and cosine transforms
• Sine integral
• Sine quadrant
• Sine wave
• Sine–Gordon equation
• Sinusoidal model
• Trigonometric functions

References

• Traupman, John C. (1966), The New College Latin & English Dictionary, Toronto: Bantam, ISBN 0-553-27619-0
• Webster's Seventh New Collegiate Dictionary, Springfield: G. & C. Merriam Company, 1969

External links

• Media related to Sine function at Wikimedia Commons

UpToDate Contents

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English Journal

• Upregulation of sine oculis homeobox homolog 3 is associated with proliferation, invasion, migration, as well as poor prognosis of esophageal cancer.

• , D D, J J, J J, .
• Anti-cancer drugs. 2019 Jul;30(6)596-603.
• Esophageal cancer (EC) is a common cancer worldwide. Sine oculis homeobox homolog (SIX3) is a human transcription factor that regulates the progression of vertebrate eye and fetal forebrain. However, studies on the function of SIX3 in human tumorigenesis remain rare. In this study, we aim to evaluat
• PMID 30672777

• A hexanucleotide repeat modifies expressivity of X-linked dystonia parkinsonism.

• Westenberger A, Reyes CJ, Saranza G, Dobricic V, Hanssen H, Domingo A, Laabs BH, Schaake S, Pozojevic J, Rakovic A, Grütz K, Begemann K, Walter U, Dressler D, Bauer P, Rolfs A, Münchau A, Kaiser FJ, Ozelius LJ, Jamora RD, Rosales RL, Diesta CCE, Lohmann K, König IR, Brüggemann N, Klein C.
• Annals of neurology. 2019 Jun;85(6)812-822.
• X-linked dystonia parkinsonism (XDP) is a neurodegenerative movement disorder caused by a single mutation: SINE-VNTR-Alu (SVA) retrotransposon insertion in TAF1. Recently, a (CCCTCT) repeat within the SVA insertion has been reported as an age-at-onset (AAO) modifier in XDP. Here we investigate the r
• PMID 30973967

• Infraslow activity as a potential modulator of corticomotor excitability.

• de Goede AA, van Putten MJAM.
• Journal of neurophysiology. 2019 May;().
• Fluctuations in cortical excitability are a candidate mechanism involved in the trial-to-trial variation of motor evoked potentials (MEPs) to transcranial magnetic stimulation (TMS). We explore whether infraslow EEG activity (<0.1 Hz) modulates corticomotor excitability by evaluating the presence of
• PMID 31116669

Japanese Journal

• Hypertranscendence of the multiple sine function for a complex period

• KATO Masaki
• Proceedings of the Japan Academy. Series. A, Mathematical sciences 95(2), 16-19, 2019-02
• NAID 40021826508

• 顔面神経麻痺重症例に対する早期減荷術の有効性

• 松見 文晶,野本 幸男
• 耳鼻咽喉科臨床 112(3), 151-156, 2019
• … Early decompression is thought to salvage facial nerve fibers from showing progression to Wallerian degeneration, and to be more beneficial than decompression after a delay.</p><p>We performed transmastoid decompression of the facial nerve for 14 patients with severe Bell's palsy, Ramsay Hunt syndrome, and zoster sine herpete within 14 days of the onset. … All the patients had poor facial motor function and ≥90% degeneration of the facial nerve fibers according to electroneurography before the surgery. …
• NAID 130007606387

• 倍精度浮動小数点正弦・余弦関数のFPGA向き計算法 (ディペンダブルコンピューティング) -- (組込み技術とネットワークに関するワークショップETNET2018)

• 豊島 悠紀夫,高木 直史
• 電子情報通信学会技術研究報告 = IEICE technical report : 信学技報 117(480), 113-117, 2018-03-07
• NAID 40021522220

Related Links

• Sine, Cosine and Tangent - Math is Fun - Maths Resources

Exercise Try this paper-based exercise where you can calculate the sine function for all angles from 0 to 360 , and then graph the result. It will help you to understand these relatively simple functions. You can also see Graphs of ...

• Sine -- from Wolfram MathWorld

The sine function sinx is one of the basic functions encountered in trigonometry (the others being the cosecant, cosine, cotangent, secant, and tangent). Let theta be an angle measured counterclockwise from the x-axis along an arc ...

• 時間のソースとして外部信号を使用し、正弦波 ...- MathWorks 日本

説明 Sine Wave Function ブロックは正弦波を出力します。このブロックは、時間ベース モードまたはサンプルベース モード ... メモ このブロックは、Math Operations ライブラリに表示される Sine Wave ブロックと同じです。ブロックのダイアログ ...

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★リンクテーブル★

リンク元	「正弦関数」
関連記事	「SINE」「function」「sine」

「正弦関数」

　　[★]

英

sine function

関

三角関数

「SINE」

　　[★]

日

短い散在反復配列、短分散型核内反復配列、短い散在性反復配列

英

short interspersed element, short interspersed nuclear element, short interspersed nucleotide element, SINEs

関

レトロトランスポゾン

quoted from http://en.wikipedia.org/wiki/Long_interspersed_nucleotide_elements#SINEs

• Short interspersed repetitive elements or Short interspersed nuclear elements[8] are short DNA sequences (<500 bases) that represent reverse-transcribed RNA molecules originally transcribed by RNA polymerase III into tRNA, rRNA, and other small nuclear RNAs. SINEs do not encode a functional reverse transcriptase protein and rely on other mobile elements for transposition. The most common SINEs in primates are called Alu sequences. Alu elements are 280 base pairs long, do not contain any coding sequences, and can be recognized by the restriction enzyme AluI (hence the name). With about 1,500,000 copies, SINEs make up about 13% of the human genome.[11] While historically viewed as "junk DNA", recent research suggests that in some rare cases both LINEs and SINEs were incorporated into novel genes, so as to evolve new functionality. The distribution of these elements has been implicated in some genetic diseases and cancers.

「function」

　　[★]

• n.

• 機能、(数学)関数、(化学)官能基

survival as a function of HbA1c in people with type 2 diabetes HbA1cの機能を考慮（指標に）した2型糖尿病の患者の生存（予後）

Figure 2A shows survival as a function of device removal during index hpspitalization, censored at 1 year or date of last contact. (Figure2Aはは最初の入院中における心臓デバイス除去に関する生存曲線で退院後1年もしくは最後の接触で打ち切っている)

• v.

• 機能する

関

functional、functional group、functionalise、functionalize、operate

　 　 　 　 　 　 　

「sine」

　　[★]

• n.

• (数)正弦、サイン

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• prep.

• ～なしに、～なく(without)

• ex.

• dermatomyositis sine myositis