# Arctangent: Value

How to find the given arctangent value: formula, 1 example and its solution.

## Formula

### Formula

Arctangent is the inverse function of tangent.

So, to solve arctangent,

set x = arctan y,
write tan x = y,
and solve the tangent equation.

x is in (-π/2 ≤ x ≤ π/2).

y = tan x is not one-to-one.
But if (-π/2 ≤ x ≤ π/2),
y = tan x is one-to-one.
So its inverse function can be defined.
This is why x is in (-π/2 ≤ x ≤ π/2).

## Example

### Solution

set x = arctan (-1).

Then tan x = -1.

x is in (-π/2 ≤ x ≤ π/2).

Draw a right triangle
that satisfies
tan x = -1 and (-π/2 ≤ x ≤ π/2).

-π/2 ≤ x ≤ π/2
So the right triangle should be in

See tan x = -1 (= -1/1).

Tangent is TOA:
Tangent,
Opposite side (-1),

So draw a right triangle on a coordinate plane
whose opposite side is -1
and whose adjacent side is 1.

Draw the angle x
that starts from the 3 o'clock position.

Find the missing side
by using the Pythagorean theorem:
12 + (-1)2 = [hypotenuse]2.

Then the hypotenuse is √2.

This is a right triangle
whose sides are 1, -1, and √2.

So this is a 45-45-90 triangle.

So the central angle is, 45º, π/4.