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Concept# Sine and cosine

Summary

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 , the sine and cosine functions are denoted simply as and .
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. They can be traced to the and functions used in Indian astronomy during the Gupta period.
Trigonometric functions#Notation
Sine and cosine are written using functional notation with the abbreviations sin and cos.
Often, if the argument is simple enough, the function value will be written without parentheses, as sin θ rather than as sin(θ).
Each of sine and cosine is a function of an angle, which is usually expressed in terms of radians or degrees. Except where explicitly stated otherwise, this article assumes that the angle is measured in radians.
To define the sine and cosine of an acute angle α, start with a right triangle that contains an angle of measure α; in the accompanying figure, angle α 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.

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