Solving Arccosine Functions

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 (0 <= x <= pi).

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 ≤ xπ.

Graphing cosine functions

Example

Find the value of the given expression. arccos (-sqrt(2)/2)

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 ≤ xπ)

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
= 3π/4.