Trigonometric function

In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. They are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to positive and negative values and even to complex numbers. All of these approaches will be presented below.

Other definitions

Theorem: There exists exactly one pair of real functions

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s, c with the following properties:

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For any x, y inmathbb{R}:

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:

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s(x)^2 + c(x)^2 = 1,,

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:s(x+y) = s(x)c(y) + c(x)s(y),,

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:c(x+y) = c(x)c(y) - s(x)s(y),,

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:0 < xc(x) < s(x) < x mathrm{for} 0 < x < 1.

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