# Solving Arccosine Functions

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

## Definition

*y* = arccos *x* is the inverse function of *y* = cos *x*.

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

is not one-to-one:

it fails the horizontal line test.

So, to define the inverse function,

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

0 ≤ *x* ≤ *π*.

Graphing cosine functions

## Example

Set *x* = arccos (-√2/2).

Next, cosine both sides.

(left side) = cos *x*

(right side) = cos (arccos [-√2/2])

= -√2/2

So cos *x* = -√2/2.

(0 ≤ *x* ≤ *π*)

To solve cos *x* = -√2/2,

draw a right triangle that satisfies

cos *x* = -√2 / 2

= -1 / √2.

Cosine: CAH.

So cos *x* = (adjacent side) / (hypotenuse) = -1 / √2.

The right triangle is a 45-45-90 triangle.

So (opposite side) = 1.

The reference angle is *π*/4. (brown)

So *x* = *π* - *π*/4

= 3*π*/4.