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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
Gupta period
Indian astronomy
Indian astronomy
(Aryabhatiya, Surya Siddhanta), via translation from Sanskrit
Sanskrit
to Arabic and then from Arabic to Latin.[1] The word "sine" comes from a Latin
Latin
mistranslation of the Arabic jiba, which is a transliteration of the Sanskrit
Sanskrit
word for half the chord, jya-ardha.[2]

Contents

1 Right-angled triangle definition 2 Unit circle
Unit circle
definition 3 Identities

3.1 Reciprocal 3.2 Inverse 3.3 Calculus 3.4 Other trigonometric functions

4 Properties relating to the quadrants 5 Series definition

5.1 Continued fraction

6 Fixed point 7 Arc length 8 Law of sines 9 Special
Special
values 10 Relationship to complex numbers

10.1 Sine
Sine
with a complex argument

10.1.1 Partial fraction and product expansions of complex sine 10.1.2 Usage of complex sine

10.2 Complex graphs

11 History

11.1 Etymology

12 Software implementations 13 See also 14 Notes 15 References 16 External links

Right-angled triangle definition[edit]

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 ⁡ α =

opposite

hypotenuse

displaystyle 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) 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
Unit circle
definition[edit]

Illustration of a unit circle. The radius has a length of 1. The variable t is an angle measure.

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 in radians.

Identities[edit] Main article: List of trigonometric identities Exact identities (using radians): These apply for all values of

θ

displaystyle theta

.

sin ⁡ ( θ ) = cos ⁡

(

π 2

− θ

)

=

1

csc ⁡ ( θ )

displaystyle sin(theta )=cos left( frac pi 2 -theta right)= frac 1 csc(theta )

Reciprocal[edit] 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 ) =

1

sin ⁡ ( A )

=

hypotenuse

opposite

=

h a

.

displaystyle csc(A)= frac 1 sin(A) = frac textrm hypotenuse textrm opposite = frac h a .

Inverse[edit]

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.

θ = arcsin ⁡

(

opposite hypotenuse

)

=

sin

− 1

(

a h

)

.

displaystyle theta =arcsin left( frac text opposite text hypotenuse right)=sin ^ -1 left( frac a h right).

k is some integer:

sin ⁡ ( y ) = x ⇔

y = arcsin ⁡ x + 2 π k ,

 or 

y = π − arcsin ⁡ ( x ) + 2 π k

displaystyle begin aligned sin(y)=xLeftrightarrow &y=arcsin x+2pi k, text or \&y=pi -arcsin(x)+2pi kend aligned

Or in one equation:

sin ⁡ ( y ) = x ⇔ y = ( − 1

)

k

arcsin ⁡ ( x ) + π k

displaystyle sin(y)=xLeftrightarrow y=(-1)^ k arcsin(x)+pi k

Arcsin satisfies:

sin ⁡ ( arcsin ⁡ ( x ) ) = x

displaystyle sin(arcsin(x))=x!

and

arcsin ⁡ ( sin ⁡ ( θ ) ) = θ

for 

π 2

≤ θ ≤

π 2

.

displaystyle arcsin(sin(theta ))=theta quad text for - frac pi 2 leq theta leq frac pi 2 .

Calculus[edit] See also: List of integrals of trigonometric functions
List of integrals of trigonometric functions
and Differentiation of trigonometric functions For the sine function:

f ( x ) = sin ⁡ ( x )

displaystyle f(x)=sin(x)

The derivative is:

f ′

( x ) = cos ⁡ ( x )

displaystyle f'(x)=cos(x)

The antiderivative is:

∫ f ( x )

d x = − cos ⁡ x + C

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

C denotes the constant of integration. Other trigonometric functions[edit]

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
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 2theta +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 4theta -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

⁡ ( θ ) +

sin

2

⁡ ( θ ) = 1

displaystyle cos ^ 2 (theta )+sin ^ 2 (theta )=1!

where sin2x means (sin(x))2. Properties relating to the quadrants[edit]

The four quadrants of a Cartesian coordinate system.

Over the four quadrants of the sine function is as follows.

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 3pi 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 3pi 2 <x<2pi

− 1 < sin ⁡ ( x ) < 0

displaystyle -1<sin(x)<0

displaystyle -

increasing convex

Points between the quadrants. k is an integer.

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

Degrees Radians

0 ≤ x < 2 π

displaystyle 0leq x<2pi

Radians

sin ⁡ ( x )

displaystyle sin(x)

Point type

0

displaystyle 0^ circ

0

displaystyle 0

2 π k

displaystyle 2pi k

0

displaystyle 0

Root, Inflection

90

displaystyle 90^ circ

π 2

displaystyle frac pi 2

2 π k +

π 2

displaystyle 2pi k+ frac pi 2

1

displaystyle 1

Maximum

180

displaystyle 180^ circ

π

displaystyle pi

2 π k − π

displaystyle 2pi k-pi

0

displaystyle 0

Root, Inflection

270

displaystyle 270^ circ

3 π

2

displaystyle frac 3pi 2

2 π k −

π 2

displaystyle 2pi k- frac pi 2

− 1

displaystyle -1

Minimum

For arguments outside those in the table, get the value using the fact the sine function has a period of 360° (or 2π rad):

sin ⁡ ( α +

360

) = sin ⁡ ( α )

displaystyle sin(alpha +360^ circ )=sin(alpha )

, or use

sin ⁡ ( α +

180

) = − sin ⁡ ( α )

displaystyle sin(alpha +180^ circ )=-sin(alpha )

. Or use

cos ⁡ ( x ) =

e

x i

+

e

− x i

2

displaystyle cos(x)= frac e^ xi +e^ -xi 2

and

sin ⁡ ( x ) =

e

x i

e

− x i

2 i

displaystyle sin(x)= frac e^ xi -e^ -xi 2i

. For complement of sine, we have

sin ⁡ (

180

− α ) = sin ⁡ ( α )

displaystyle sin(180^ circ -alpha )=sin(alpha )

. Series definition[edit]

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
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

( 4 n + k )

⁡ ( 0 ) =

0

when 

k = 0

1

when 

k = 1

0

when 

k = 2

− 1

when 

k = 3

displaystyle 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=3end cases

This gives the following Taylor series
Taylor series
expansion at x = 0. One can then use the theory of Taylor series
Taylor series
to show that the following identities hold for all real numbers x (where x is the angle in radians) :[3]

sin ⁡ x

= x −

x

3

3 !

+

x

5

5 !

x

7

7 !

+ ⋯

=

n = 0

( − 1

)

n

( 2 n + 1 ) !

x

2 n + 1

displaystyle 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 messy factors involving powers of π/180: if x is the number of degrees, the number of radians is y = πx /180, so

sin ⁡

x

d e g

= sin ⁡

y

r a d

=

π 180

x −

(

π 180

)

3

x

3

3 !

+

(

π 180

)

5

x

5

5 !

(

π 180

)

7

x

7

7 !

+ ⋯ .

displaystyle 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

sin ⁡ 0 = 0

 and 

sin ⁡

2 x

= 2 sin ⁡ x cos ⁡ x

cos

2

⁡ x +

sin

2

⁡ x = 1

 and 

cos ⁡

2 x

=

cos

2

⁡ x −

sin

2

⁡ x

displaystyle begin aligned sin 0=0& text and sin 2x =2sin xcos 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 ≈ x

 when 

x ≈ 0.

displaystyle sin xapprox x text when xapprox 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[edit] The sine function can also be represented as a generalized continued fraction:

sin ⁡ x =

x

1 +

x

2

2 ⋅ 3 −

x

2

+

2 ⋅ 3

x

2

4 ⋅ 5 −

x

2

+

4 ⋅ 5

x

2

6 ⋅ 7 −

x

2

+ ⋱

.

displaystyle sin x= cfrac x 1+ cfrac x^ 2 2cdot 3-x^ 2 + cfrac 2cdot 3x^ 2 4cdot 5-x^ 2 + cfrac 4cdot 5x^ 2 6cdot 7-x^ 2 +ddots .

The continued fraction representation expresses the real number values, both rational and irrational, of the sine function. Fixed point[edit]

The fixed point iteration xn+1 = sin xn 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[edit] The arc length of the sine curve between

a

displaystyle a

and

b

displaystyle b

is

a

b

1 + cos ⁡ ( x

)

2

d x

displaystyle int _ a ^ b ! sqrt 1+cos(x)^ 2 ,dx

This integral is an elliptic integral of the second kind. The arc length for a full period is

4

π

3

/

2

2

Γ ( 1

/

4

)

2

+

Γ ( 1

/

4

)

2

2 π

= 7.640395578 …

displaystyle frac 4pi ^ 3/2 sqrt 2 Gamma (1/4)^ 2 + frac Gamma (1/4)^ 2 sqrt 2pi =7.640395578ldots

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 2pi

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

π

displaystyle pi

. The Fourier series
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

1.21600672 × x + 0.10317093 sin ⁡ ( 2 x ) − 0.00220445 sin ⁡ ( 4 x ) + 0.00012584 sin ⁡ ( 6 x ) − 0.00001011 sin ⁡ ( 8 x ) + ⋯

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

Law of sines[edit] 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:

sin ⁡ A

a

=

sin ⁡ B

b

=

sin ⁡ C

c

.

displaystyle frac sin A a = frac sin B b = frac sin C c .

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

a

sin ⁡ A

=

b

sin ⁡ B

=

c

sin ⁡ C

= 2 R ,

displaystyle 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
Special
values[edit] 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 ⁡

π 60

= sin ⁡

3

=

( 2 −

12

)

5 +

5

+ (

10

2

) (

3

+ 1 )

16

displaystyle sin frac pi 60 =sin 3^ circ = frac (2- sqrt 12 ) sqrt 5+ sqrt 5 +( sqrt 10 - sqrt 2 )( sqrt 3 +1) 16

 A019812

sin ⁡

π 30

= sin ⁡

6

=

30 −

180

5

− 1

8

displaystyle sin frac pi 30 =sin 6^ circ = frac sqrt 30- sqrt 180 - sqrt 5 -1 8

 A019815

sin ⁡

π 20

= sin ⁡

9

=

10

+

2

20 −

80

8

displaystyle sin frac pi 20 =sin 9^ circ = frac sqrt 10 + sqrt 2 - sqrt 20- sqrt 80 8

 A019818

sin ⁡

π 15

= sin ⁡

12

=

10 +

20

+

3

15

8

displaystyle sin frac pi 15 =sin 12^ circ = frac sqrt 10+ sqrt 20 + sqrt 3 - sqrt 15 8

 A019821

sin ⁡

π 10

= sin ⁡

18

=

5

− 1

4

=

1 2

φ

− 1

displaystyle sin frac pi 10 =sin 18^ circ = frac sqrt 5 -1 4 = tfrac 1 2 varphi ^ -1

 A019827

sin ⁡

7 π

60

= sin ⁡

21

=

( 2 +

12

)

5 −

5

− (

10

+

2

) (

3

− 1 )

16

displaystyle sin frac 7pi 60 =sin 21^ circ = frac (2+ sqrt 12 ) sqrt 5- sqrt 5 -( sqrt 10 + sqrt 2 )( sqrt 3 -1) 16

 A019830

sin ⁡

π 8

= sin ⁡

22.5

=

2 −

2

2

displaystyle sin frac pi 8 =sin 22.5^ circ = frac sqrt 2- sqrt 2 2

sin ⁡

2 π

15

= sin ⁡

24

=

3

+

15

10 −

20

8

displaystyle sin frac 2pi 15 =sin 24^ circ = frac sqrt 3 + sqrt 15 - sqrt 10- sqrt 20 8

 A019833

sin ⁡

3 π

20

= sin ⁡

27

=

20 +

80

10

+

2

8

displaystyle sin frac 3pi 20 =sin 27^ circ = frac sqrt 20+ sqrt 80 - sqrt 10 + sqrt 2 8

 A019836

sin ⁡

11 π

60

= sin ⁡

33

=

(

12

− 2 )

5 +

5

+ (

10

2

) (

3

+ 1 )

16

displaystyle sin frac 11pi 60 =sin 33^ circ = frac ( sqrt 12 -2) sqrt 5+ sqrt 5 +( sqrt 10 - sqrt 2 )( sqrt 3 +1) 16

 A019842

sin ⁡

π 5

= sin ⁡

36

=

10 −

20

4

displaystyle sin frac pi 5 =sin 36^ circ = frac sqrt 10- sqrt 20 4

 A019845

sin ⁡

13 π

60

= sin ⁡

39

=

( 2 −

12

)

5 −

5

+ (

10

+

2

) (

3

+ 1 )

16

displaystyle sin frac 13pi 60 =sin 39^ circ = frac (2- sqrt 12 ) sqrt 5- sqrt 5 +( sqrt 10 + sqrt 2 )( sqrt 3 +1) 16

 A019848

sin ⁡

7 π

30

= sin ⁡

42

=

30 +

180

5

+ 1

8

displaystyle sin frac 7pi 30 =sin 42^ circ = frac sqrt 30+ sqrt 180 - sqrt 5 +1 8

 A019851

Relationship to complex numbers[edit] Main article: Trigonometric functions
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
Sine
is used to determine the imaginary part of a complex number given in polar coordinates (r,φ):

z = r ( cos ⁡ φ + i sin ⁡ φ )

displaystyle z=r(cos varphi +isin varphi )

the imaginary part is:

Im ⁡ ( z ) = r sin ⁡ φ

displaystyle operatorname Im (z)=rsin 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
Sine
can also take a complex number as an argument.

Sine
Sine
with a complex argument[edit]

sin ⁡ z

displaystyle sin z

Domain coloring
Domain coloring
of sin(z) over (-π,π) on x and y axes. 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:

sin ⁡ z

=

n = 0

( − 1

)

n

( 2 n + 1 ) !

z

2 n + 1

=

e

i z

e

− i z

2 i

=

sinh ⁡

(

i z

)

i

displaystyle 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(izright) i end aligned

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

sin ⁡ x = Im ⁡ (

e

i x

) .

displaystyle sin x=operatorname Im (e^ ix ).

For purely imaginary numbers:

sin ⁡ i y = i sinh ⁡ y .

displaystyle sin iy=isinh y.

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

sin ⁡ ( x + i y )

= sin ⁡ x cos ⁡ i y + cos ⁡ x sin ⁡ i y

= sin ⁡ x cosh ⁡ y + i cos ⁡ x sinh ⁡ y .

displaystyle begin aligned sin(x+iy)&=sin xcos iy+cos xsin iy\&=sin xcosh y+icos xsinh y.end aligned

Partial fraction and product expansions of complex sine[edit] Using the partial fraction expansion technique in complex analysis, one can find that the infinite series

n = − ∞

( − 1

)

n

z − n

=

1 z

− 2 z

n = 1

( − 1

)

n

n

2

z

2

displaystyle begin aligned sum _ n=-infty ^ infty frac (-1)^ n z-n = frac 1 z -2zsum _ n=1 ^ infty frac (-1)^ n n^ 2 -z^ 2 end aligned

both converge and are equal to

π

sin ⁡ π z

displaystyle frac pi sin pi z

. Similarly, one can show that

π

2

sin

2

⁡ ( π z )

=

n = − ∞

1

( z − n

)

2

.

displaystyle 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

sin ⁡ π z = π z

n = 1

(

1 −

z

2

n

2

)

.

displaystyle begin aligned sin pi z=pi zprod _ n=1 ^ infty Bigl ( 1- frac z^ 2 n^ 2 Bigr ) .end aligned

Usage of complex sine[edit] sin z is found in the functional equation for the Gamma function,

Γ ( s ) Γ ( 1 − s ) =

π

sin ⁡ π s

,

displaystyle Gamma (s)Gamma (1-s)= pi over sin pi s ,

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

ζ ( s ) = 2 ( 2 π

)

s − 1

Γ ( 1 − s ) sin ⁡ ( π s

/

2 ) ζ ( 1 − s ) .

displaystyle zeta (s)=2(2pi )^ 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:

Δ u (

x

1

,

x

2

) = 0.

displaystyle Delta u(x_ 1 ,x_ 2 )=0.

It is also related with level curves of pendulum.[4] Complex graphs[edit]

Sine
Sine
function in the complex plane

real component imaginary component magnitude

Arcsine function in the complex plane

real component imaginary component magnitude

History[edit] 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
Hipparchus
of Nicaea (180–125 BCE) and Ptolemy
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
Gupta period
(320 to 550 CE) Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit
Sanskrit
to Arabic and then from Arabic to Latin.[1] 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.[5] Roger Cotes computed the derivative of sine in his Harmonia Mensurarum (1722).[6] 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.[7] Etymology[edit]

Look up sine in Wiktionary, the free dictionary.

Etymologically, the word sine derives from the Sanskrit
Sanskrit
word for chord, jiva*(jya being its more popular synonym). This was transliterated in Arabic as jiba جــيــب, abbreviated jb جــــب . Since Arabic is written without short vowels, "jb" was interpreted as the word jaib جــيــب, which means "bosom", when the Arabic text was translated in the 12th century into Latin
Latin
by Gerard of Cremona. Gerard used the Latin
Latin
equivalent for "bosom", sinus (which means "bosom" or "bay" or "fold").[8][9] The English form sine was introduced in the 1590s. Software implementations[edit] 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.[10] 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.[citation needed] The CORDIC
CORDIC
algorithm is commonly used in scientific calculators. See also[edit]

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
Polar sine
— a generalization to vertex angles Proofs of trigonometric identities Sinc function Sine
Sine
and cosine transforms Sine
Sine
integral Sine
Sine
quadrant Sine
Sine
wave Sine–Gordon equation Sinusoidal model Trigonometric functions Trigonometry
Trigonometry
in Galois fields

Notes[edit]

^ a b Uta C. Merzbach, Carl B. Boyer (2011), A History of Mathematics, Hoboken, N.J.: John Wiley & Sons, 3rd ed., p. 189. ^ Victor J. Katz (2008), A History of Mathematics, Boston: Addison-Wesley, 3rd. ed., p. 253, sidebar 8.1. "Archived copy" (PDF). Archived (PDF) from the original on 2015-04-14. Retrieved 2015-04-09.  ^ See Ahlfors, pages 43–44. ^ math.stackexchange questions : why-are-the-phase-portrait-of-the-simple-plane-pendulum-and-a-domain-coloring-of ... Archived 2014-03-30 at the Wayback Machine. ^ Nicolás Bourbaki (1994). Elements of the History of Mathematics. Springer.  ^ "Why the sine has a simple derivative Archived 2011-07-20 at the Wayback Machine.", in Historical Notes for Calculus
Calculus
Teachers Archived 2011-07-20 at the Wayback Machine. by V. Frederick Rickey Archived 2011-07-20 at the Wayback Machine. ^ See Merzbach, Boyer (2011). ^ Eli Maor (1998), Trigonometric Delights, Princeton: Princeton University Press, p. 35-36. ^ Victor J. Katz (2008), A History of Mathematics, Boston: Addison-Wesley, 3rd. ed., p. 253, sidebar 8.1. "Archived copy" (PDF). Archived (PDF) from the original on 2015-04-14. Retrieved 2015-04-09.  ^ Grand Challenges of Informatics, Paul Zimmermann. September 20, 2006 – p. 14/31 "Archived copy" (PDF). Archived (PDF) from the original on 2011-07-16. Retrieved 2010-09-11. 

References[edit]

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

External links[edit]

Media related to Sine
Sine
function at

.