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Cosine is a trigonometric ratio. In a right triangle with an angle $ \theta $ ,

$ \cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}} $

$ \text{Adjacent} $ is the side of the triangle that is part of angle $ \theta $ , and $ \text{hypotenuse} $ is the side opposite the right angle.


Properties

The cosine of an angle is the x-coordinate of the point of intersection of said angle and a unit circle.

As a result of Euler's formula, the cosine function can also be represented as

$ \cos(\theta)=\frac{e^{i\theta}+e^{-i\theta}}{2} $

The reciprocal of cosine is secant (abbreviated as $ \sec $), while its inverse is $ \arccos $ or $ \cos^{-1} $ . Note that sine is not being raised to the power of -1; this is an inverse function, not a reciprocal.

The derivative of $ \cos(x) $ is $ -\sin(x) $ , while its antiderivative is $ \sin(x) $ .

The trigonometric addition/subtraction identiy of cosine is:

$ \cos(a\pm b)=\cos(a)\cos(b)\mp\sin(a)\sin(b) $

The square of cosine:

$ \cos^2(\theta)=\frac{\cos(2\theta)+1}{2} $

Imaginary number:

$ \cos(ix)=\cosh(x) $

Limits

$ \lim_{x\to 0} \frac{1 - cos x}{x} = 0 $

Power series

More formally defined, the cosine is defined by the power series, which is convergent for all $ x $,

$ \cos(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} $

Where $ n! $ denotes the factorial of n. From this, the above properties may be derived.

See also