Trigonometric function

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In mathematics there are 6 trigonometric functions: sine, cosine, tangent, cotangent, secant and cosecant. Secant and cosecant are rarely used.

You can define them using geometry. Sometimes people define them using power series:

\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots = \sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)!}
\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots = \sum_{n=0}^\infty \frac{(-1)^nx^{2n}}{(2n)!}
\tan x = \frac{\sin x}{\cos x}
\operatorname{cotan} x = \frac{\cos x}{\sin x}
\sec x = \frac{1}{\cos x}
\operatorname{cosec} x = \frac{1}{\sin x}.

Some important identities:

sin2x + cos2x = 1.
\sin \left(x+y\right)=\sin x \cos y + \cos x \sin y
\sin \left(x-y\right)=\sin x \cos y - \cos x \sin y
\cos \left(x+y\right)=\cos x \cos y - \sin x \sin y
\cos \left(x-y\right)=\cos x \cos y + \sin x \sin y

See also: trigonometry