# Inverse Tan

In trigonometry, inverse tan is the inverse function of the trigonometric function tangent. It is used to calculate the angle by applying the tangent ratio of the angle, opposite divided by adjacent. It is naming convention for all inverse trigonometric functions is to use the prefix ‘arc’ and hence inverse tangent is denoted by arctan. Although it is not uncommon to use $tan^{-1}x$, we will use arctan throughout this article.

## Properties of Inverse Tan Function

The basic properties of inverse tan function, arctan, are listed below:

Notation: y = arctan (x)

Defined as: x = tan (y)

Domain of the ratio: all real numbers

Range of the principal value in radians: $-\frac{\pi}{2}< y < \frac{\pi}{2}$

Range of the principal value in degrees: $-90^{\circ}< y < 90^{\circ}$

## Relationship Between Inverse Tangent Function and Other Trigonometric Functions

Consider a triangle whose length of adjacent and opposite are 1 and x respectively. Therefore the length of the hypotenuse is $\sqrt{1+x^{2}}$. For this triangle, if the angle $\Theta$ is the arctan, then the following relationships hold true for the three basic trigonometric functions:

Sin (arctan(x)) = $\frac{x}{\sqrt{1+x^{2}}}$

Cos (arctan(x)) = $\frac{1}{\sqrt{1+x^{2}}}$

Tan (arctan(x)) = x

## Inverse Tan Formula for Addition

The formula for adding inverse tan is derived from tan addition formula

$tan (a\pm b) = \frac{tan a \pm tan b}{1\mp tan(a)tan(b)}$

In this formula, by putting a = arctan x and b = arctan y, we get

$arctan(x)\pm arctan(y)= \frac{x\pm y}{1\mp xy}$ $(mod \pi )$

Where $xy \neq 1$

## Calculus of Arctan Function

This section gives the formula to calculate the derivative and integral of the arctan function.

### Derivative

The derivative of arctan x is denoted by $\frac{d}{dx}arctan(x)$ and for complex values of x , the derivative is equal to $\frac{1}{1+x^{2}}$ for $x \neq -i,+i$

### Integral

For obtaining an expression for the definite integral of the inverse tan function, the derivative is integrated and the value at one point is fixed. The expression is:

$arctan (x) = \int_{0}^{x}\frac{1}{y^{2}+1}dy$

## Inverse Tan Formula for Integration

Some of the important formulae for calculating integrals of expressions involving the arctan function are:

$\int arctan (x)dx = x arctan (x)- \frac{In (x^{2}+1)}{2}+C$

$\int arctan (ax)dx = x arctan (ax)- \frac{In (a^{2}x^{2}+1)}{2a}+C$

$\int x arctan (ax)dx = \frac{x^{2} arctan (ax)}{2}- \frac{arctan(ax)}{2a^{2}}- \frac{x}{2a}+C$

$\int x^{2} arctan (ax)dx = \frac{x^{3} arctan (ax)}{3}- \frac{In (a^{2}x^{2}+1)}{6a^{3}}- \frac{x^{2}}{6a}+C$