# Sine and cosine

**Sine and cosine** are the two fundamental trigonometric functions: for a real number \( t \), the sine of \( t \) is the y-coordinate and the cosine of \( t \) is the x-coordinate of the point where an arc of length \( t \) on the unit circle ends.<sup>[1](https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle)</sup> Written \( \sin t \) and \( \cos t \), they take any real number as input, always return a value between −1 and 1, and repeat their values every \( 2\pi \). They relate the angles of a triangle to its side lengths, describe rotation and oscillation, and serve as building blocks for [Fourier analysis](https://www.edgepedia.org/fourier-analysis), differential equations, and numerical computation.<sup>[2](https://openstax.org/books/calculus-volume-1/pages/1-3-trigonometric-functions)</sup>

| Key fact | Detail |
|---|---|
| Unit-circle definition | For a central angle \( t \) on a circle of radius 1 centered at the origin, \( \cos t = x \) and \( \sin t = y \), the coordinates of the arc's endpoint<sup>[1](https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle)</sup> |
| Domain and range | Both functions accept any real input and return values in \( [-1, 1] \)<sup>[1](https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle)</sup> |
| Pythagorean identity | \( \cos^2 t + \sin^2 t = 1 \), obtained by substituting \( x = \cos t \), \( y = \sin t \) into the unit-circle equation \( x^2 + y^2 = 1 \)<sup>[1](https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle)</sup> |
| Periodicity | \( \sin(z + 2k\pi) = \sin z \) and \( \cos(z + 2k\pi) = \cos z \) for any integer \( k \)<sup>[3](https://dlmf.nist.gov/4.14)</sup> |
| Derivatives | \( \frac{d}{dz}\sin z = \cos z \) and \( \frac{d}{dz}\cos z = -\sin z \)<sup>[4](https://dlmf.nist.gov/4.20)</sup> |
| Complex form | \( \sin z = \frac{e^{iz} - e^{-iz}}{2i} \), \( \cos z = \frac{e^{iz} + e^{-iz}}{2} \)<sup>[3](https://dlmf.nist.gov/4.14)</sup> |
| Historical origin | Aryabhata (476–550) tabulated 24 sine values at intervals of 3°45′ in the *Aryabhatiya* (499 CE), the first known table of half-chords<sup>[5](https://mathshistory.st-andrews.ac.uk//Biographies/Aryabhata_I/)</sup><sup> • </sup><sup>[6](https://mathshistory.st-andrews.ac.uk/Strick/aryabhata.pdf)</sup> |

## Definition on the unit circle

A unit circle has its center at \( (0,0) \) and radius 1, so the length of an intercepted arc equals the radian measure of its central angle \( t \). If \( (x, y) \) is the endpoint on the circle of an arc of length \( t \), then \( \cos t = x \) and \( \sin t = y \). The input of each function is the angle measure; the output is a coordinate of the corresponding point.<sup>[1](https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle)</sup>

Because the definitions use arc length rather than a triangle, they extend to every real number, including angles beyond a full turn and negative angles. There is no lower or upper limit on the inputs, and since both coordinates of a point on the circle lie between −1 and 1, the range of both functions is \( [-1, 1] \).<sup>[1](https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle)</sup> The same circle gives the Pythagorean identity \( \cos^2 t + \sin^2 t = 1 \), one of the most used relations in mathematics.<sup>[1](https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle)</sup>

The notation has conventions worth knowing: \( \sin t \) means the same as \( \sin(t) \), and \( \cos^2 t \) is shorthand for \( (\cos(t))^2 \). Many calculators and computers do not recognize the shorthand, so extra parentheses are advised when entering calculations.<sup>[1](https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle)</sup>

## Relation to right triangles

The unit-circle definition generalizes the older right-triangle definition. For an angle \( \theta \) in standard position with its terminal side meeting a circle at \( P = (x, y) \), the six trigonometric functions are defined in terms of the coordinates \( x \) and \( y \); trigonometric functions let us use angle measures in radians or degrees to find coordinates of a point on any circle, not only a unit circle, or to find an angle given a point on a circle.<sup>[2](https://openstax.org/books/calculus-volume-1/pages/1-3-trigonometric-functions)</sup> In a right triangle, sine and cosine are ratios of side lengths, and the two settings agree because the unit circle's radius serves as the hypotenuse. The two definitions are equivalent: a published analysis shows how the geometrical definitions, using right triangles and unit circles, connect to the differential definition through elementary geometry and calculus, arriving at the harmonic differential equations.<sup>[7](https://arxiv.org/abs/1110.6106)</sup>

## Analysis: series, derivatives, and the complex exponential

In complex analysis the functions are defined algebraically from the exponential:

\[ \sin z = \frac{e^{iz} - e^{-iz}}{2i}, \qquad \cos z = \frac{e^{iz} + e^{-iz}}{2}. \]

These formulas give \( e^{\pm iz} = \cos z \pm i \sin z \), Euler's relation between the exponential and the circular functions.<sup>[3](https://dlmf.nist.gov/4.14)</sup> Both \( \sin z \) and \( \cos z \) are entire functions, meaning they are analytic everywhere in the [complex plane](https://www.edgepedia.org/complex-plane). Their zeros are exactly \( z = k\pi \) for sine and \( z = (k + \tfrac{1}{2})\pi \) for cosine, with \( k \) an integer.<sup>[3](https://dlmf.nist.gov/4.14)</sup>

Their Maclaurin series converge for every \( z \):<sup>[8](https://dlmf.nist.gov/4.19)</sup>

\[ \sin z = z - \frac{z^3}{3!} + \frac{z^5}{5!} - \frac{z^7}{7!} + \cdots, \qquad \cos z = 1 - \frac{z^2}{2!} + \frac{z^4}{4!} - \frac{z^6}{6!} + \cdots. \]

Differentiation cycles the two functions with a sign change: \( \frac{d}{dz}\sin z = \cos z \) and \( \frac{d}{dz}\cos z = -\sin z \).<sup>[4](https://dlmf.nist.gov/4.20)</sup> This cycling makes sine and cosine the natural solutions of the harmonic oscillator equation \( w'' + a^2 w = 0 \), whose general solution is \( w = A\cos(az) + B\sin(az) \) for constants \( A \) and \( B \).<sup>[4](https://dlmf.nist.gov/4.20)</sup> That equation models vibration, alternating current, and any system with a restoring force proportional to displacement, which is why the two functions appear throughout physics and engineering.<sup>[9](https://aeresources.gatech.edu/num_meth/Webpage/FFT/Theory/Theory.php)</sup>

## History

The functions descend from chord tables used in astronomy. Aryabhata (476–550), working at Kusumapura in India, gave a table of sines at intervals of \( \frac{90°}{24} = 3°45′ \), computed with a recurrence formula for \( \sin(n+1)x - \sin nx \) in terms of \( \sin nx \) and \( \sin(n-1)x \); he also introduced the versine, \( 1 - \cos \), into trigonometry.<sup>[5](https://mathshistory.st-andrews.ac.uk//Biographies/Aryabhata_I/)</sup> His *Aryabhatiya* lists 24 sine values, 225, 224, 222, 219, 215, 210, 205, 199, 191, 183, 174, 164, 154, 143, 131, 119, 106, 93, 79, 65, 51, 37, 22, 7, measured in parts of a radius of 3438, a radius chosen so that each minute of arc corresponds to one unit of arc length.<sup>[6](https://mathshistory.st-andrews.ac.uk/Strick/aryabhata.pdf)</sup> Unlike the ancient Greeks, who tabulated full chords, [Aryabhata](https://www.edgepedia.org/aryabhata) tabulated half-chords, calling them *ardha-jya*, or *jya* for short.<sup>[6](https://mathshistory.st-andrews.ac.uk/Strick/aryabhata.pdf)</sup>

The modern word "sine" is the result of a translation chain. In Arabic translation *jya* became *jiba*, a word without meaning; when Gerhard of Cremona translated the Toledan tables of Al-Zarqali into Latin, he confused *jiba* with the Arabic word *jaib*, which translates as *sinus*.<sup>[6](https://mathshistory.st-andrews.ac.uk/Strick/aryabhata.pdf)</sup> Once *sinus* was in use, mathematicians coined a companion term for the complementary function: early Latin names included *sinus rectus complementi* ([Regiomontanus](https://www.edgepedia.org/regiomontanus), c. 1463) and *sinus rectus secundus*, used by Peter Apianus in 1541.<sup>[10](https://mathshistory.st-andrews.ac.uk/Miller/mathword/c/)</sup> The English mathematician Edmund Gunter (1581–1626) suggested the term *co.sinus* in his 1620 *Canon triangulorum*, and [John Newton](https://www.edgepedia.org/john-newton) modified it to *cosinus* in 1658, the form that gained general favor; Gunter also invented the word *cotangent* in the same work.<sup>[10](https://mathshistory.st-andrews.ac.uk/Miller/mathword/c/)</sup><sup> • </sup><sup>[11](https://mathshistory.st-andrews.ac.uk/Biographies/Gunter/)</sup>

Indian mathematicians continued refining the tables, since accurate sines were needed to compute planetary positions. Aryabhata II (c. 920–1000) constructed a sine table correct to five decimal places when measured in decimal parts of the radius, in his *Mahasiddhanta*.<sup>[12](https://mathshistory.st-andrews.ac.uk/Biographies/Aryabhata_II/)</sup> In Kerala, Madhava (1350–1425) and his school developed series expansions of the sine and cosine; in the Indian convention the sine of an angle meant \( r \sin \theta \) with \( r \) the radius, and the *[Yuktibhasa](https://www.edgepedia.org/yuktibhasa)* (c. 1500 CE) describes evaluations and expansions of trigonometric functions.<sup>[13](https://mathshistory.st-andrews.ac.uk/Biographies/Madhava/)</sup><sup> • </sup><sup>[14](https://arxiv.org/abs/2304.03684)</sup>

## Computation

Modern libraries compute sine and cosine from the complex exponential definitions or by polynomial and rational approximations after reducing the argument to a small interval.<sup>[3](https://dlmf.nist.gov/4.14)</sup><sup> • </sup><sup>[15](https://dlmf.nist.gov/4.45)</sup> On hardware without a floating-point unit, the CORDIC algorithm is a common alternative: it uses only shifts and additions, and fourteen iterations yield both sine and cosine of an angle between 0 and \( \pi/2 \) accurate to about \( \pm 0.000488 \); larger angles are handled by domain reduction. A drawback is that the algorithm computes both quantities even when only one is wanted.<sup>[16](https://nvlpubs.nist.gov/nistpubs/Legacy/IR/nistir4480.pdf)</sup> The IEEE 754-2019 standard, which specifies floating-point formats and operations for computer systems, underpins the arithmetic environments in which such library functions are implemented.<sup>[17](https://standards.ieee.org/ieee/754/6210/)</sup>

## Applications

Periodic phenomena are modeled with sinusoidal functions. In aerospace engineering, vibration, periodic loading of rotor or wind turbine blades, and active control data are modeled in the time domain using sinusoidal functions, characterized by amplitude (the height of the oscillation over one period), phase (the lag of the response behind the forcing), and frequency, with angular frequency \( \omega = 2\pi f \) and period \( T = 1/f \).<sup>[9](https://aeresources.gatech.edu/num_meth/Webpage/FFT/Theory/Theory.php)</sup> Because any periodic signal can be decomposed into sines and cosines, the two functions are the basis of the [Fourier transform](https://www.edgepedia.org/fourier-transform), which underlies spectral analysis across physics and engineering.<sup>[18](https://arxiv.org/abs/1811.05331)</sup>

## References

1. [7.3 Unit Circle, Algebra and Trigonometry 2e, OpenStax](https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle)
2. [1.3 Trigonometric Functions, Calculus Volume 1, OpenStax](https://openstax.org/books/calculus-volume-1/pages/1-3-trigonometric-functions)
3. [DLMF §4.14 Definitions and Periodicity, NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/4.14)
4. [DLMF §4.20 Derivatives and Differential Equations, NIST DLMF](https://dlmf.nist.gov/4.20)
5. [Aryabhata (476–550), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk//Biographies/Aryabhata_I/)
6. [Aryabhata, MacTutor History of Mathematics (Strick biographical PDF)](https://mathshistory.st-andrews.ac.uk/Strick/aryabhata.pdf)
7. [Correspondence between geometrical and differential definitions of the sine and cosine functions, arXiv:1110.6106](https://arxiv.org/abs/1110.6106)
8. [DLMF §4.19 Maclaurin Series and Laurent Series, NIST DLMF](https://dlmf.nist.gov/4.19)
9. [Fast Fourier Transforms: Theory, Georgia Tech AEResources](https://aeresources.gatech.edu/num_meth/Webpage/FFT/Theory/Theory.php)
10. [Earliest Known Uses of Some of the Words of Mathematics (C), MacTutor](https://mathshistory.st-andrews.ac.uk/Miller/mathword/c/)
11. [Edmund Gunter (1581–1626), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Gunter/)
12. [Aryabhata II (920–1000), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Aryabhata_II/)
13. [Madhava (1350–1425), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Madhava/)
14. [India and the Calculus of Trigonometric Functions, arXiv:2304.03684](https://arxiv.org/abs/2304.03684)
15. [DLMF §4.45 Methods of Computation, NIST DLMF](https://dlmf.nist.gov/4.45)
16. [Assembly code to compute sine and cosine using the CORDIC algorithm, NISTIR 4480](https://nvlpubs.nist.gov/nistpubs/Legacy/IR/nistir4480.pdf)
17. [IEEE 754-2019, IEEE Standard for Floating-Point Arithmetic](https://standards.ieee.org/ieee/754/6210/)
18. [Applications of sinusoidal phase modulation in temporal optics, arXiv:1811.05331](https://arxiv.org/abs/1811.05331)

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