# Solving Arctangent Functions

How to solve arctangent function problems: definition, example, and its solution.

## Definition

*y* = arctan *x* is the inverse function of *y* = tan *x*.

Of course the graph of *y* = tan *x*

is not one-to-one:

it fails the horizontal line test.

So, to define the inverse function,

the domain of *y* = tan *x* is fixed like this:

-*π*/2 < *x* < *π*/2.

Graphing tangent functions

## Example

Set *x* = arctan (-√3).

Next, tangent both sides.

(left side) = tan *x*

(right side) = tan (arctan [-√3])

= -√3

So tan *x* = -√3.

(-*π*/2 < *x* < *π*/2)

To solve tan *x* = -√3,

draw a right triangle that satisfies

tan *x* = -√3 / 1.

Tangent: TOA.

So tan *x* = (opposite side) / (adjacent side) = -√3 / 1.

The right triangle is a 30-60-90 triangle.

So (hypotenuse) = 2.

So arctan (-√3) = *x*

= -*π*/3.

So cos (arctan (-√3)) = cos *x*

= cos (-*π*/3).

Then, from the right triangle,

cos (-*π*/3) = 1/2.

Actually, you don't need to write *x* = -*π*/3

for finding cos *x*,

because from the right triangle,

you can directly find cos *x* = 1/2.

Just wanted to show you

arctan (-√3) = -*π*/3.