Sine - wikipedia Sine For other uses, see Sine ( disambiguation ). Not to be confused with sign. Sine Basic features Parity odd Domain ( − ∞, ∞ ) Codomain ( − 1, 1 ) Period 2π Specific values At zero 0 Maxima ( ( 2k + 1⁄2 ) π, 1 ) Minima ( ( 2k − 1⁄2 ) π, − 1 ) Specific features Root kπ Critical point kπ − π / 2 Inflection point kπ Fixed point 0 For real numbers. 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 '' 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 ⁡ 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 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 ⁡ ( 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 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 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 ). 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 ) = x Leftrightarrow &y = arcsin x + 2 pi k, ( \text ( or ) ) &y = pi - arcsin ( x ) + 2 pi k end ( aligned ) ) ) Or in one equation : sin ⁡ ( y ) = x ⇔ y = ( − 1 ) k arcsin ⁡ ( x ) + π 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 ⁡ ( 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 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 ⁡ ( 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 = 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 ⁡ ( θ ) + sin 2 ⁡ ( θ ) = 1 ( displaystyle cos ^ ( 2 ) ( \theta ) + sin ^ ( 2 ) ( \theta ) = 1 ! ) where sin x means ( sin ( x ) ). 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 ∘ < 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 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 0 leq x < 2 pi ) Radians sin ⁡ ( x ) ( displaystyle sin ( x ) ) Point type 0 ∘ ( displaystyle 0 ^ ( circ ) ) 0 ( displaystyle 0 ) 2 π k ( displaystyle 2 pi k ) 0 ( displaystyle 0 ) Root, Inflection 90 ∘ ( displaystyle 90 ^ ( circ ) ) π 2 ( displaystyle ( \frac ( pi ) ( 2 ) ) ) 2 π k + π 2 ( displaystyle 2 pi k+ ( \frac ( pi ) ( 2 ) ) ) 1 ( displaystyle 1 ) Maximum 180 ∘ ( displaystyle 180 ^ ( circ ) ) π ( displaystyle pi ) 2 π k − π ( displaystyle 2 pi k - pi ) 0 ( displaystyle 0 ) Root, Inflection 270 ∘ ( displaystyle 270 ^ ( circ ) ) 3 π 2 ( displaystyle ( \frac ( 3 pi ) ( 2 ) ) ) 2 π k − π 2 ( displaystyle 2 pi 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 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& ( \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 ) : 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 ) = 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 ≈ x when x ≈ 0. ( displaystyle 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 = 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 ) ) ( 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 expresses the real number values, both rational and irrational, of the sine function. Fixed point The fixed point iteration x = sin x with initial value x = 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 π 3 / 2 Γ ( 1 / 4 ) 2 + Γ ( 1 / 4 ) 2 2 π = 7.640395578... ( displaystyle ( \frac ( 4 ( sqrt ( 2 ) ) , pi ^ ( 3 / 2 ) ) ( 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 1.21600672 × 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 ) - 0.00220445 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 : 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 0 0 0 0 180 ° π 200 1 / 2 15 ° 1 / 12 π 16 2 / 3 1 / 24 6 − 2 4 ( displaystyle ( \frac ( ( sqrt ( 6 ) ) - ( sqrt ( 2 ) ) ) ( 4 ) ) ) 0.258819045102521 165 ° 11 / 12 π 183 1 / 3 11 / 24 30 ° 1 / 6π 33 1 / 3 1 / 12 1 / 2 0.5 150 ° 5 / 6π 166 2 / 3 5 / 12 45 ° 1 / 4π 50 1 / 8 2 2 ( displaystyle ( \frac ( sqrt ( 2 ) ) ( 2 ) ) ) 0.707106781186548 135 ° 3 / 4π 150 3 / 8 60 ° 1 / 3π 66 2 / 3 1 / 6 3 2 ( displaystyle ( \frac ( sqrt ( 3 ) ) ( 2 ) ) ) 0.866025403784439 120 ° 2 / 3π 133 1 / 3 1 / 3 75 ° 5 / 12 π 83 1 / 3 5 / 24 6 + 2 4 ( displaystyle ( \frac ( ( sqrt ( 6 ) ) + ( sqrt ( 2 ) ) ) ( 4 ) ) ) 0.965925826289068 105 ° 7 / 12 π 116 2 / 3 7 / 24 90 ° 1 / 2π 100 1 / 4 90 degree increments : x in degrees 0 ° 90 ° 180 ° 270 ° 360 ° x in radians 0 π / 2 π 3π / 2 2π x in gons 0 100 200 300 400 x in turns 0 1 / 4 1 / 2 3 / 4 sin x 0 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 ( 7 pi ) ( 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 ( 2 pi ) ( 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 ( 3 pi ) ( 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 ( 11 pi ) ( 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 ( 13 pi ) ( 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 ( 7 pi ) ( 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 ⁡ φ + i sin ⁡ φ ) ( displaystyle z = r ( cos \varphi + i sin \varphi ) , ) the imaginary part is : Im ⁡ ( z ) = r sin ⁡ φ ( displaystyle 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 ) 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 ( iz \right ) ) ( i ) ) end ( aligned ) ) ) where i = − 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 = 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 : 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 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 ∑ 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 ) ) - 2z sum _ ( 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 we can find 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 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, 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 ( 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 : Δ 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 ( 10 ). 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 Sinc function Sine and cosine transforms Sine integral Sine quadrant Sine wave Sine -- Gordon equation Sinusoidal model Trigonometric functions Trigonometry in Galois fields Notes ^ Jump up to : Uta C. Merzbach, Carl B. Boyer ( 2011 ), A History of Mathematics, Hoboken, N.J. : John Wiley & Sons, 3rd ed., p. 189. Jump up ^ Victor J. Katz ( 2008 ), A History of Mathematics, Boston : Addison - Wesley, 3rd. ed., p. 253, sidebar 8.1. ( 1 ) Jump up ^ See Ahlfors, pages 43 -- 44. Jump up ^ math. stackexchange questions : why - are - the - phase - portrait - of - the - simple - plane - pendulum - and - a-domain - coloring - of... Jump up ^ Nicolás Bourbaki ( 1994 ). Elements of the History of Mathematics. Springer. Jump up ^ `` Why the sine has a simple derivative '', in Historical Notes for Calculus Teachers by V. Frederick Rickey Jump up ^ See Merzbach, Boyer ( 2011 ). Jump up ^ Eli Maor ( 1998 ), Trigonometric Delights, Princeton : Princeton University Press, p. 35 - 36. Jump up ^ Victor J. Katz ( 2008 ), A History of Mathematics, Boston : Addison - Wesley, 3rd. ed., p. 253, sidebar 8.1. ( 2 ) Jump up ^ Grand Challenges of Informatics, Paul Zimmermann. September 20, 2006 -- p. 14 / 31 ( 3 ) 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 Retrieved from `` https://en.wikipedia.org/w/index.php?title=Sine&oldid=808924347 '' Categories : Trigonometric functions Hidden categories : All articles with unsourced statements Articles with unsourced statements from October 2012 Talk Contents About Wikipedia Afrikaans Azərbaycanca Bân - lâm - gú Български Bosanski Čeština Corsu Dansk Deutsch Eesti Ελληνικά Español Esperanto Euskara فارسی Français Galego हिन्दी Hrvatski Bahasa Indonesia Italiano עברית ქართული Latina Latviešu Македонски മലയാളം मराठी Norsk Norsk nynorsk Oʻzbekcha / ўзбекча ਪੰਜਾਬੀ Piemontèis Português Ripoarisch Română संस्कृतम् Scots සිංහල Slovenčina کوردی Српски / srpski Srpskohrvatski / српскохрватски Basa Sunda Svenska தமிழ் Türkçe Українська اردو Tiếng Việt 粵語 中文 Edit links This page was last edited on 6 November 2017, at 01 : 51. 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Question: where does sine cosine and tangent come from
Answer: the jyā and koṭi-jyā functions used in Gupta period Indian astronomy