# 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

### 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

either quadrant I or quadrant IV.

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

Tangent is TOA:

Tangent,

Opposite side (-1),

Adjacent 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:

1^{2} + (-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.

Radian Measure

So π/4 is the answer.