Knowlpedia

Definition

Cosine function

The even trigonometric function defined by its entire power series.

The cosine function is

cos⁡x=∑n=0∞(−1)nx2n(2n)!.\cos x=\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{(2n)!}.

The converges for all real xx, is even, and gives cos⁡0=1\cos0=1.

Derivative

Termwise differentiation gives cos⁡′x=−sin⁡x\cos' x=-\sin x, where is given by its companion odd series.

Rotation

The addition formulas imply that

(cos⁡θ−sin⁡θsin⁡θcos⁡θ)\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}

is an orthogonal matrix, and multiplying the matrices for two angles adds their angles. Together with sine, cosine has period 2π2\pi; this fixes the radian angle convention used in polar and cylindrical coordinates.

References