Cosine is one of the most important trigonometric functions. Cosine of an angle is defined in right-angled triangle. Let us consider a right angled triangle ABC as shown in the given figure:
Cosine Graph
Cosine of angle $\theta$ is given by the following formula:

Cosine Formula
The values of cosine of few important angles are given below:
cos 0$^o$ 1
cos 30$^o$
$\frac{\sqrt{3}}{2}$
cos 45$^o$ $\frac{1}{\sqrt{2}}$
cos 60$^o$
$\frac{1}{2}$
cos 90$^o$
0

Important properties of a cosine function are as follows:
  • Cosine function is continuous.
  • It is an even function that implies that cos (-x) = cos x.
  • Cosine function is periodic with period 2$\pi$.
  • Domain of cosine is (-$\infty$, +$\infty$) and range is (-1, 1).
  • cos n$\pi$ = (-1)n, where n is an integer.
  • Derivative: $\frac{\mathrm{d} }{\mathrm{d} x}$$\cos x = -\sin x$
  • Integral: $\int \cos x = \sin x+c$, where c is a constant.

Few important formulas of cosi
ne are as follows:
  1. cos (A + B) = cos A cos B - sin A sin B
  2. cos (A - B) = cos A cos B + sin A sin B
  3. cos2A = cos2 A - sin2 A = 1 - 2sin2 A = 2cos2 A - 1

Cosine law:

  • $\cos A$ = $\frac{b^{2} + c^{2} - a^{2}}{2bc}$

  • $\cos B$ = $\frac{c^{2} + a^{2} - b^{2}}{2ac}$

  • $\cos C$ = $\frac{a^{2} + b^{2} - c^{2}}{2ab}$
Where, A, B and C are angles of a triangle and a, b and c are the sides opposite to them respectively.
Graph of cosine is shown below. Here, $\theta$ is measured in radians.

Graph of cosine
Inverse of cosine is denoted by cos-1$\theta$ or arc(cos $\theta$). It is the inverse function of cosine. $\cos^{-1}\theta \neq$ $\frac{1}{\cos \theta }$cos is not a one-one function. So, its domain must be restricted from 0 to $\pi$ for its inverse function. Graph of inverse cosine looks as shown in the figure below:
Inverse Cosine Graph
Graph of negative cosine or -cosine is demonstrated as shown below:
Negative Cosine Graph
If we compare this graph with positive cosine graph, we can find that negative cosine is a reflected image of positive cosine.