## Definitions

### General

The inverse cosine function, in modern notation written as arccos(x), gives the angle θ, so that:

\[\cos \theta = x \]

Because the values of cosine function range between -1 and 1, the domain of argument x, in arccos function, is restricted to the same range: [-1,1]. Also, due to the periodical nature of the cosine function, there are many angles θ that can give the same cosine value (i.e. θ+2π, θ+4π, etc.). As a result, it is impossible to define a single inverse function, unless the range of the return values is restricted, so that a one-to-one relationship between θ and cosθ can be established. Therefore, multiple branches of the arccos function can be defined. Commonly, the desired range of θ values spans between 0 and π. The branch of arccos, in that case, is called the principal branch.

### Series

The arccos function can be defined in a Taylor series form, like this:

\[ \begin{split} \arccos x & = \frac{\pi}{2}-\arcsin x = \\ & = \frac{\pi}{2} -\sum_{n=0}^{\infty}\frac{\binom{2n}{n}x^{2n+1}}{4^n \left(2n+1\right)} = \\ & = \frac{\pi}{2} -x - \frac{x^3}{6} - \frac{3x^5}{40} - \frac{5x^7}{112} \cdots \end{split} \]

The above series is valid for |x|≤1. From the expanded form of the series, it can be seen that the higher terms become insignificant, for values of x close to zero, resulting in the following quite useful approximation:

\[ \arccos x \approx \frac{\pi}{2} -x, \quad x \to 0, \quad \textrm{result x in radians} \]

### Properties

The derivative of the arccos function is:

\[ \left(\arccos x\right)' = -\frac{1}{\sqrt{1-x^2}}, \quad |x|\ne 1 \]

The integral of the arccos function is given by:

\[ \int \arccos x\, \mathrm{d}x = x \arccos x - \sqrt{1-x^2} + C \]

The following properties are also valid for the arccos function:

\[ \begin{split} & \cos (\arccos x) &= x \\ \\ & \sin (\arccos x) &= \sqrt{1-x^2}\\ \\ & \tan (\arccos x) &= \frac{\sqrt{1-x^2}}{x} \\ \\ & \arccos \left(-x\right) &= \pi -\arccos x \\ \\ & \arccos x &= \frac{\pi}{2} - \arcsin x \\ \\ \end{split} \]