Arctangent

What Is the Arctangent?

The arctangent is the inverse function of the tangent. In other words, it allows you to determine an angle when you already know its tangent value. The function is commonly written as arctan, arctg, tan^-1, or tg^-1.

$$ y = \arctan x $$

where x is a tangent value.

The result of the arctangent function is the angle y whose tangent is equal to x.

$$ y = \arctan x $$

Equivalently,

$$ x = \tan y $$

This relationship makes the arctangent one of the most useful inverse trigonometric functions, especially when solving geometric and trigonometric problems.

The graph of the arctangent function is shown below.

Note: The domain of the arctangent function is the set of all real numbers because the tangent function can take any real value on its principal interval.

$$ D_{\arctan} = (-\infty,+\infty) $$

The range of the arctangent is

$$ \left(-\frac{\pi}{2},\frac{\pi}{2}\right) $$

which is the interval used to define the principal values of the inverse tangent function.

A Practical Example

A simple example helps illustrate how the arctangent works.

The tangent of π/4 is equal to 1.

$$ \tan \frac{\pi}{4} = 1 $$

Therefore, the angle whose tangent is 1 must be π/4.

$$ \arctan 1 = \frac{\pi}{4} $$

Since the arctangent reverses the action of the tangent function, applying one after the other returns the original angle, provided the angle lies within the principal interval.

$$ \arctan \left( \tan \frac{\pi}{4} \right) = \frac{\pi}{4} $$

Note: The equality above holds because π/4 belongs to the interval (-π/2, π/2), which is the domain used for the inverse tangent function.

How to Draw the Graph of the Arctangent

The graph of the arctangent can be obtained directly from the graph of the tangent function.

Start with the graph of y = tan x.

The tangent function is not invertible over its entire domain because different angles can have the same tangent value.

To make the function invertible, we restrict its domain to the interval (-π/2, π/2).

Within this interval, the tangent function is strictly increasing and takes every real value exactly once. As a result, it becomes a bijection and therefore has an inverse.

A general property of inverse functions is that their graphs are mirror images of each other across the line y = x.

Reflect the restricted tangent graph across this line.

The reflected curve is the graph of the arctangent function.

The arctangent function is itself invertible.

Its inverse is the tangent function restricted to the interval (-π/2, π/2).

Note: The tangent function may also be restricted to other intervals on which it is one-to-one, such as (π/2, 3π/2). In that case, the corresponding inverse function returns angles in the chosen interval rather than in the principal interval of the standard arctangent.

The same approach can be used to construct inverse functions from any bijective restriction of the tangent function.