## Definitions

### General

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

\[\tan \theta = x \]

Due to the periodical nature of the tangent function, there are many angles θ that can give the same tangent value (i.e. θ+π, θ+3π, 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 tanθ can be established. Therefore, multiple branches of the arctan function can be defined. Commonly, the desired range of θ values spans between -π/2 and π/2. The branch of arctan, in that case, is called the principal branch.

### Series

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

\[ \begin{split} \arctan x & = \sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{2n+1} = \\ & = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} \cdots \end{split} \]

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:

\[ \arctan x \approx x, \quad x \to 0, \quad \textrm{result x in radians} \]

### Properties

The derivative of the arctan function is:

\[ \left(\arctan x\right)' = \frac{1}{1+x^2} \]

The integral of the arctan function is given by:

\[ \int \arctan x\, \mathrm{d}x = x \arctan x -\frac{\ln\left(1+x^2\right)}{2} + C \]

The following properties are also valid for the arctan function:

\[ \begin{split} & \tan (\arctan x) &= x \\ \\ & \sin (\arctan x) &= \frac{x}{\sqrt{1+x^2}} \\ \\ & \cos (\arctan x) &= \frac{1}{\sqrt{1+x^2}} \\ \\ & \arctan \left(-x\right) &=- \arctan x \\ \\ & \arctan x &= \frac{\pi}{2} - \textrm{arccot}\, x \\ \\ \end{split} \]