Sine

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:

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

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 ${\displaystyle \sin(\alpha )}$ appears to depend on the choice of right triangle containing an angle of measure α. However, this is not the case: all such triangles are similar, and so the ratio is the same for each of them.

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

Unit circle: a circle with radius one

Let a line through the origin intersect the unit circle, making an angle of θ with the positive half of the x-axis. The x- and y-coordinates of this point of intersection are equal to cos(θ) and sin(θ), respectively. This definition is consistent with the right-angled triangle definition of sine and cosine when 0° < θ < 90°. Because the length of the hypotenuse of the unit circle is always 1, ${\displaystyle \sin(\theta )={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}={\frac {\textrm {opposite}}{1}}={\textrm {opposite}}}$. The length of the opposite side of the triangle is simply the y-coordinate. A similar argument can be made for the cosine function to show that cos(θ) ${\displaystyle ={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}}$ when 0° < θ < 90°, even under the new definition using the unit circle. tan(θ) is then defined as ${\displaystyle {\frac {\sin(\theta )}{\cos(\theta )}}}$, or, equivalently, as the gradient of the line segment.

Using the unit circle definition has the advantage that the angle can take any real argument. This can also be achieved by requiring certain symmetries and that sine be a periodic function.

• 
  Animation showing how the sine function (in red) ${\displaystyle y=\sin(\theta )}$ is graphed from the y-coordinate (red dot) of a point on the unit circle (in green) at an angle of θ.

Exact identities (using radians):

These apply for all values of ${\displaystyle \theta }$.

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

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.

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

k is some integer:

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

Or in one equation:

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

Arcsin satisfies:

${\displaystyle \sin(\arcsin(x))=x\!}$

and

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

Other trigonometric functions

The sine and cosine functions are related in multiple ways. The two functions are out of phase by 90°: ${\displaystyle \sin(\pi /2-x)}$ = ${\displaystyle \cos(x)}$ 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	${\displaystyle \sin(\theta )}$	${\displaystyle =\pm {\sqrt {1-\cos ^{2}(\theta )}}}$	+	+	−	−	${\displaystyle =\operatorname {sgn} \left(\cos \left(\theta -{\frac {\pi }{2}}\right)\right){\sqrt {1-\cos ^{2}(\theta )}}}$
	${\displaystyle \cos(\theta )}$	${\displaystyle =\pm {\sqrt {1-\sin ^{2}(\theta )}}}$	+	−	−	+	${\displaystyle =\operatorname {sgn} \left(\sin \left(\theta +{\frac {\pi }{2}}\right)\right){\sqrt {1-\sin ^{2}(\theta )}}}$
cot	${\displaystyle \sin(\theta )}$	${\displaystyle =\pm {\frac {1}{\sqrt {1+\cot ^{2}(\theta )}}}}$	+	+	−	−	${\displaystyle =\operatorname {sgn} \left(\cot \left({\frac {\theta }{2}}\right)\right){\frac {1}{\sqrt {1+\cot ^{2}(\theta )}}}}$
	${\displaystyle \cot(\theta )}$	${\displaystyle =\pm {\frac {\sqrt {1-\sin ^{2}(\theta )}}{\sin(\theta )}}}$	+	−	−	+	${\displaystyle =\operatorname {sgn} \left(\sin \left(\theta +{\frac {\pi }{2}}\right)\right){\frac {\sqrt {1-\sin ^{2}(\theta )}}{\sin(\theta )}}}$
tan	${\displaystyle \sin(\theta )}$	${\displaystyle =\pm {\frac {\tan(\theta )}{\sqrt {1+\tan ^{2}(\theta )}}}}$	+	−	−	+	${\displaystyle =\operatorname {sgn} \left(\tan \left({\frac {2\theta +\pi }{4}}\right)\right){\frac {\tan(\theta )}{\sqrt {1+\tan ^{2}(\theta )}}}}$
	${\displaystyle \tan(\theta )}$	${\displaystyle =\pm {\frac {\sin(\theta )}{\sqrt {1-\sin ^{2}(\theta )}}}}$	+	−	−	+	${\displaystyle =\operatorname {sgn} \left(\sin \left(\theta +{\frac {\pi }{2}}\right)\right){\frac {\sin(\theta )}{\sqrt {1-\sin ^{2}(\theta )}}}}$
sec	${\displaystyle \sin(\theta )}$	${\displaystyle =\pm {\frac {\sqrt {\sec ^{2}(\theta )-1}}{\sec(\theta )}}}$	+	−	+	−	${\displaystyle =\operatorname {sgn} \left(\sec \left({\frac {4\theta -\pi }{2}}\right)\right){\frac {\sqrt {\sec ^{2}(\theta )-1}}{\sec(\theta )}}}$
	${\displaystyle \sec(\theta )}$	${\displaystyle =\pm {\frac {1}{\sqrt {1-\sin ^{2}(\theta )}}}}$	+	−	−	+	${\displaystyle =\operatorname {sgn} \left(\sin \left(\theta +{\frac {\pi }{2}}\right)\right){\frac {1}{\sqrt {1-\sin ^{2}(\theta )}}}}$

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:

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

where sin²(x) means (sin(x))².

Sine squared function

Sine function in blue and sine squared function in red. The Y axis is in radians.

The graph shows both the sine function and the sine squared function, with the sine in blue and sine squared in red. Both graphs have the same shape, but with different ranges of values, and different periods. Sine squared has only positive values, but twice the number of periods.

The sine squared function can be expressed as a modified sine wave from the Pythagorean identity and power reduction by the cosine double-angle formula:[3]

${\displaystyle \sin ^{2}(\theta )\ ={\frac {1-\sin(2\theta +{\tfrac {\pi }{2}})}{2}}\ }$

The four quadrants of a Cartesian coordinate system.

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

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

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

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

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

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:

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

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) :[4]

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

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

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

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

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

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

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

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.

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

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

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

The arc length for a full period is ${\textstyle {\frac {4{\sqrt {2\pi ^{3}}}}{\Gamma (1/4)^{2}}}+{\frac {\Gamma (1/4)^{2}}{\sqrt {2\pi }}}=7.640395578\ldots }$ where ${\displaystyle \Gamma }$ is the gamma function.

The arc length of the sine curve from 0 to x is the above number divided by ${\displaystyle 2\pi }$ 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

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

The leading term in the above equation, and the limit of arc length to distance ratio is given by:

${\displaystyle {\frac {\pi ^{2}+2\Gamma \left({\frac {3}{4}}\right)^{4}}{{\sqrt {2}}\pi ^{3/2}\Gamma \left({\frac {3}{4}}\right)^{2}}}}$

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:

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

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

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

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.

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	${\displaystyle {\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}$	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	${\displaystyle {\frac {\sqrt {2}}{2}}}$	0.707106781186548
135°	3/4π	150g	3/8
60°	1/3π	66+2/3g	1/6	${\displaystyle {\frac {\sqrt {3}}{2}}}$	0.866025403784439
120°	2/3π	133+1/3g	1/3
75°	5/12π	83+1/3g	5/24	${\displaystyle {\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}$	0.965925826289068
105°	7/12π	116+2/3g	7/24
90°	1/2π	100g	1/4	1	1

90 degree increments:

Other values not listed above:

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

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

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

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

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

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

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

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

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

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

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

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

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

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):

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

the imaginary part is:

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

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

${\displaystyle \sin(z)}$

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

sin(z) as a vector field

${\displaystyle \sin(\theta )}$ is the imaginary part of ${\displaystyle e^{i\theta }}$.

The definition of the sine function for complex arguments z:

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

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

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

For purely imaginary numbers:

${\displaystyle \sin(iy)=i\sinh(y).}$

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

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

Complex graphs

Sine function in the complex plane

real component	imaginary component	magnitude

Arcsine function in the complex plane

real component	imaginary component	magnitude

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 of sine and versine (1 - 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.[1]

All six trigonometric functions in current use were known in Islamic mathematics by the 9th century, as was the law of sines, used in solving triangles.[6] With the exception of the sine (which was adopted from Indian mathematics), the other five modern trigonometric functions were discovered by Arabic mathematicians, including the cosine, tangent, cotangent, secant and cosecant.[6] Al-Khwārizmī (c. 780–850) produced tables of sines, cosines and tangents.[7][8] Muhammad ibn Jābir al-Harrānī al-Battānī (853–929) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°.[8]

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.[9] Roger Cotes computed the derivative of sine in his Harmonia Mensurarum (1722).[10] 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.[11]

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 جيب, which however is meaningless in that language and abbreviated jb جب . Since Arabic is written without short vowels, "jb" was interpreted as the word jaib جيب, which means "bosom". When the Arabic texts were translated in the 12th century into Latin by Gerard of Cremona, he used the Latin equivalent for "bosom", sinus (which means "bosom" or "bay" or "fold").[12][13] Gerard was probably not the first scholar to use this translation; Robert of Chester appears to have preceded him and there is evidence of even earlier usage.[14] The English form sine was introduced in the 1590s.

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.[15] 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.

• Traupman, Ph.D., 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

• Media related to Sine function at Wikimedia Commons

Look up sine in Wiktionary, the free dictionary.