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


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


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


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:


Imaginary number:



\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