# Sine and cosine

In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine 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), and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle ${\displaystyle \theta }$, the sine and cosine functions are denoted simply as ${\displaystyle \sin \theta }$ and ${\displaystyle \cos \theta }$.[1]

Sine and cosine
General information
General definition{\displaystyle {\begin{aligned}&\sin(\alpha )={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}\\[8pt]&\cos(\alpha )={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}\\[8pt]\end{aligned}}}
Fields of applicationTrigonometry, fourier series, etc.

More generally, the definitions of sine and cosine can be extended to any real value in terms of the lengths of certain line segments in a unit circle. More modern definitions express the sine and cosine as infinite series, or as the solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers.

The sine and cosine functions are 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 functions sine and cosine can be traced to the functions jyā and koṭi-jyā, used in Indian astronomy during the Gupta period (Aryabhatiya and Surya Siddhanta), via translation from Sanskrit to Arabic, and then from Arabic to Latin.[2] The word sine (Latin sinus) comes from a Latin mistranslation by Robert of Chester of the Arabic jiba, itself a transliteration of the Sanskrit word for half of a chord, jya-ardha.[3] The word cosine derives from a contraction of the medieval Latin complementi sinus.[4]

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