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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 Indian astronomy (_ Aryabhatiya _, _ Surya Siddhanta _), via translation from Sanskrit to Arabic and then from Arabic to Latin. The word "sine" comes from a Latin mistranslation of the Arabic _jiba_, which is a transliteration of the Sanskrit word for half the chord, _jya-ardha_.

CONTENTS

* 1 Right-angled triangle definition * 2 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 values

* 10 Relationship to complex numbers

* 10.1 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

_ 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 = 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 DEFINITION

_ 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 ( ) {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

Main article: List of trigonometric identities

Exact identities (using radians ):

These apply for all values of {displaystyle theta } . sin ( ) = cos ( 2 ) = 1 csc ( ) {displaystyle 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 ) = 1 sin ( A ) = hypotenuse opposite = h a . {displaystyle 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 . = 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 k , or y = arcsin ( x ) + 2 k {displaystyle {begin{aligned}sin(y)=x Leftrightarrow &y=arcsin x+2pi k,{text{ or }}\ width:40.121ex; height:6.176ex;" alt="{begin{aligned}sin(y)=x Leftrightarrow &y=arcsin x+2pi k,{text{ or }}\"> sin ( y ) = x y = ( 1 ) k arcsin ( x ) + k {displaystyle sin(y)=x Leftrightarrow y=(-1)^{k}arcsin(x)+pi k}

Arcsin satisfies: sin ( arcsin ( x ) ) = x {displaystyle sin(arcsin(x))=x!}

and arcsin ( sin ( ) ) = for 2 2 . {displaystyle 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 ) {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

_ The sine and cosine functions are related in multiple ways. The two functions are out of phase by 90°: sin ( / 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 ( ) {displaystyle sin(theta )} = 1 cos 2 ( ) {displaystyle =pm {sqrt {1-cos ^{2}(theta )}}} + + − − = sgn ( cos ( 2 ) ) 1 cos 2 ( ) {displaystyle =operatorname {sgn} left(cos left(theta -{frac {pi }{2}}right)right){sqrt {1-cos ^{2}(theta )}}}

cos ( ) {displaystyle cos(theta )} = 1 sin 2 ( ) {displaystyle =pm {sqrt {1-sin ^{2}(theta )}}} + − − + = sgn ( sin ( + 2 ) ) 1 sin 2 ( ) {displaystyle =operatorname {sgn} left(sin left(theta +{frac {pi }{2}}right)right){sqrt {1-sin ^{2}(theta )}}}

COT sin ( ) {displaystyle sin(theta )} = 1 1 + cot 2 ( ) {displaystyle =pm {frac {1}{sqrt {1+cot ^{2}(theta )}}}} + + − − = sgn ( cot ( 2 ) ) 1 1 + cot 2 ( ) {displaystyle =operatorname {sgn} left(cot left({frac {theta }{2}}right)right){frac {1}{sqrt {1+cot ^{2}(theta )}}}}

cot ( ) {displaystyle cot(theta )} = 1 sin 2 ( ) sin ( ) {displaystyle =pm {frac {sqrt {1-sin ^{2}(theta )}}{sin(theta )}}} + − − + = sgn ( sin ( + 2 ) ) 1 sin 2 ( ) sin ( ) {displaystyle =operatorname {sgn} left(sin left(theta +{frac {pi }{2}}right)right){frac {sqrt {1-sin ^{2}(theta )}}{sin(theta )}}}

TAN sin ( ) {displaystyle sin(theta )} = tan ( ) 1 + tan 2 ( ) {displaystyle =pm {frac {tan(theta )}{sqrt {1+tan ^{2}(theta )}}}} + − − + = sgn ( tan ( 2 + 4 ) ) tan ( ) 1 + tan 2 ( ) {displaystyle =operatorname {sgn} left(tan left({frac {2theta +pi }{4}}right)right){frac {tan(theta )}{sqrt {1+tan ^{2}(theta )}}}}

tan ( ) {displaystyle tan(theta )} = sin ( ) 1 sin 2 ( ) {displaystyle =pm {frac {sin(theta )}{sqrt {1-sin ^{2}(theta )}}}} + − − + = sgn ( sin ( + 2 ) ) sin ( ) 1 sin 2 ( ) {displaystyle =operatorname {sgn} left(sin left(theta +{frac {pi }{2}}right)right){frac {sin(theta )}{sqrt {1-sin ^{2}(theta )}}}}

SEC sin ( ) {displaystyle sin(theta )} = sec 2 ( ) 1 sec ( ) {displaystyle =pm {frac {sqrt {sec ^{2}(theta )-1}}{sec(theta )}}} + − + − = sgn ( sec ( 4 2 ) ) sec 2 ( ) 1 sec ( ) {displaystyle =operatorname {sgn} left(sec left({frac {4theta -pi }{2}}right)right){frac {sqrt {sec ^{2}(theta )-1}}{sec(theta )}}}

sec ( ) {displaystyle sec(theta )} = 1 1 sin 2 ( ) {displaystyle =pm {frac {1}{sqrt {1-sin ^{2}(theta )}}}} + − − + = sgn ( sin ( + 2 ) ) 1 1 sin 2 ( ) {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 ( ) + sin 2 ( ) = 1 {displaystyle 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.

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

QUADRANT DEGREES RADIANS VALUE SIGN MONOTONY CONVEXITY

1st Quadrant 0 2 {displaystyle 0 2 ( x ) {displaystyle 180^{circ } 2 {displaystyle pi ( x ) 2 1 {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 ( X ) {DISPLAYSTYLE SIN(X)} POINT TYPE

0 {displaystyle 0^{circ }} 0 {displaystyle 0} 2 k {displaystyle 2pi k} 0 {displaystyle 0} Root , Inflection

90 {displaystyle 90^{circ }} 2 {displaystyle {frac {pi }{2}}} 2 k + 2 {displaystyle 2pi k+{frac {pi }{2}}} 1 {displaystyle 1} Maximum

180 {displaystyle 180^{circ }} {displaystyle pi } 2 k {displaystyle 2pi k-pi } 0 {displaystyle 0} Root , Inflection

270 {displaystyle 270^{circ }} 3 2 {displaystyle {frac {3pi }{2}}} 2 k 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 ( + 360 ) = sin ( ) {displaystyle sin(alpha +360^{circ })=sin(alpha )} , or use sin ( + 180 ) = sin ( ) {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 ) = sin ( ) {displaystyle sin(180^{circ }-alpha )=sin(alpha )} .

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 ( 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 width:34.062ex; height:11.176ex;" alt="sin ^{(4n+k)}(0)={begin{cases}0&{text{when }}k=0\1&{text{when }}k=1\0&{text{when }}k=2\-1"> 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 \ width:33.665ex; height:14.843ex;" alt="{begin{aligned}sin x&=x-{frac {x^{3}}{3!}}+{frac {x^{5}}{5!}}-{frac {x^{7}}{7!}}+cdots \"> sin x d e g = sin y r a d = 180 x ( 180 ) 3 x 3 3 ! + ( 180 ) 5 x 5 5 ! ( 180 ) 7 x 7 7 ! + . {displaystyle {begin{aligned}sin x_{mathrm {deg} }&=sin y_{mathrm {rad} }\ margin-bottom: -0.278ex; width:68.166ex; height:9.176ex;" alt="{begin{aligned}sin x_{mathrm {deg} }&=sin y_{mathrm {rad} }\"> 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 width:48.428ex; height:6.176ex;" alt="{begin{aligned}sin 0=0&{text{ and }}sin {2x}=2sin xcos x\cos ^{2}x+sin ^{2}x=1"> 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

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

_ The fixed point iteration x__n_+1 = sin _x__n_ with initial value _x_0 = 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 {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 2 3 / 2 ( 1 / 4 ) 2 + ( 1 / 4 ) 2 2 = 7.640395578 {displaystyle {frac {4{sqrt {2}},pi ^{3/2}}{Gamma (1/4)^{2}}}+{frac {Gamma (1/4)^{2}}{sqrt {2pi }}}=7.640395578ldots } where {displaystyle Gamma } is the Gamma function .

The arc length of the sine curve from _0_ to _x_ is the above number divided by 2 {displaystyle 2pi } _ times x_, plus a correction that varies periodically in _x_ with period {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

1.21600672 x + 0.10317093 sin ( 2 x ) 0.00220445 sin ( 4 x ) + 0.00012584 sin ( 6 x ) 0.00001011 sin ( 8 x ) + {displaystyle 1.21600672,times ,x,+,0.10317093,sin(2x)-0.00220445sin(4x)+0.00012584sin(6x)-0.00001011sin(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_: 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 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 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 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 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 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 10 = sin 18 = 5 1 4 = 1 2 1 {displaystyle sin {frac {pi }{10}}=sin 18^{circ }={frac {{sqrt {5}}-1}{4}}={tfrac {1}{2}}varphi ^{-1},}  A019827 sin 7 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 8 = sin 22.5 = 2 2 2 , {displaystyle sin {frac {pi }{8}}=sin 22.5^{circ }={frac {sqrt {2-{sqrt {2}}}}{2}},} sin 2 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 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 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 5 = sin 36 = 10 20 4 {displaystyle sin {frac {pi }{5}}=sin 36^{circ }={frac {sqrt {10-{sqrt {20}}}}{4}},}  A019845 sin 13 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 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

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 + i sin ) {displaystyle z=r(cos varphi +isin varphi ),}

the imaginary part is: Im ( z ) = r sin {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 can also take a complex number as an argument.

SINE WITH A COMPLEX ARGUMENT

sin z {displaystyle sin z,}

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 ( ) {displaystyle sin(theta )} is the imaginary part of e i {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}},\ width:27.182ex; height:18.843ex;" alt="{begin{aligned}sin z&=sum _{n=0}^{infty }{frac {(-1)^{n}}{(2n+1)!}}z^{2n+1}\&={frac {e^{iz}-e^{-iz}}{2i}},\"> 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\ width:41.455ex; height:6.176ex;" alt="{begin{aligned}sin(x+iy)&=sin xcos iy+cos xsin iy\"> 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 sin z {displaystyle {frac {pi }{sin pi z}}} .

Similarly we can find 2 sin 2 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 z = 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

_sin z_ is found in the functional equation for the Gamma function , ( s ) ( 1 s ) = sin 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 , ( s ) = 2 ( 2 ) s 1 ( 1 s ) sin ( s / 2 ) ( 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 : u ( x 1 , x 2 ) = 0. {displaystyle 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

_ Look up SINE _ in Wiktionary, the free dictionary.

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_ 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 by Gerard of Cremona . Gerard used the Latin equivalent for "bosom", _sinus_ (which means "bosom" or "bay" or "fold"). 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.

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 * Sine and cosine transforms * Sine quadrant * Sine wave * Sine–Gordon equation * Sinusoidal model * Trigonometric functions * Trigonometry in Galois fields

NOTES

* ^ _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. * ^ See Ahlfors, pages 43–44. * ^ math.stackexchange questions : why-are-the-phase-portrait-of-the-simple-plane-pendulum-and-a-domain-coloring-of ... * ^ Nicolás Bourbaki (1994). _Elements of the History of Mathematics_. Springer. * ^ "Why the sine has a simple derivative", in _Historical Notes for Calculus Teachers_ by V. Frederick Rickey * ^ 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. * ^ Grand Challenges of Informatics, Paul Zimmermann. September 20, 2006 – p. 14/31

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

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