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 koṭijyā functions
used in
Gupta period
Contents 1 Rightangled triangle definition
2
Unit circle
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
10.1
Sine
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 Rightangled 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 rightangled 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
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 xaxis, intersect the unit circle. The x and ycoordinates 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 ycoordinate (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 (sin1). As sine is noninjective, 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
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 /2x) = 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
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 1cos ^ 2 (theta ) + + − − = sgn ( cos ( θ − π 2 ) ) 1 − cos 2 ( θ ) displaystyle =operatorname sgn left(cos left(theta  frac pi 2 right)right) sqrt 1cos ^ 2 (theta ) cos ( θ ) displaystyle cos(theta ) = ± 1 − sin 2 ( θ ) displaystyle =pm sqrt 1sin ^ 2 (theta ) + − − + = sgn ( sin ( θ + π 2 ) ) 1 − sin 2 ( θ ) displaystyle =operatorname sgn left(sin left(theta + frac pi 2 right)right) sqrt 1sin ^ 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 1sin ^ 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 1sin ^ 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 1sin ^ 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 1sin ^ 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 1sin ^ 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 1sin ^ 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 kpi 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
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 + kth 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
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 xleft( 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 xsin ^ 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 3x^ 2 + cfrac 2cdot 3x^ 2 4cdot 5x^ 2 + cfrac 4cdot 5x^ 2 6cdot 7x^ 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
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
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
An illustration of the complex plane. The imaginary numbers are on the vertical coordinate axis.
Sine
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
sin z displaystyle sin z
Domain coloring
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 zn = 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 (zn)^ 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 (1s)= pi over sin pi s , which in turn is found in the functional equation for the Riemann zetafunction, ζ ( s ) = 2 ( 2 π ) s − 1 Γ ( 1 − s ) sin ( π s / 2 ) ζ ( 1 − s ) . displaystyle zeta (s)=2(2pi )^ s1 Gamma (1s)sin(pi s/2)zeta (1s). 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
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
Look up sine in Wiktionary, the free dictionary. Etymologically, the word sine derives from the
Sanskrit
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
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:
AddisonWesley, 3rd. ed., p. 253, sidebar 8.1. "Archived copy" (PDF).
Archived (PDF) from the original on 20150414. Retrieved
20150409.
^ See Ahlfors, pages 43–44.
^ math.stackexchange questions :
whyarethephaseportraitofthesimpleplanependulumandadomaincoloringof
... Archived 20140330 at the Wayback Machine.
^ Nicolás Bourbaki (1994). Elements of the History of Mathematics.
Springer.
^ "Why the sine has a simple derivative Archived 20110720 at the
Wayback Machine.", in Historical Notes for
Calculus
References[edit] Traupman, John C. (1966), The New College
Latin
External links[edit] Media related to
Sine
