# Arccosine: Value

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

## Formula

Arccosine is the inverse function of cosine.

So, to solve arccosine,

set x = arccos y,

write cos x = y,

and solve the cosine equation.

x is in (0 ≤ x ≤ π).

y = cos x is not one-to-one.

But if (0 ≤ x ≤ π),

y = cos x is one-to-one.

So its inverse function can be defined.

This is why x is in (0 ≤ x ≤ π).

## Examplearccos (-1/2)

set x = arccos (-1/2).

Then cos x = -1/2.

x is in (0 ≤ x ≤ π).

Draw a right triangle

that satisfies

cos x = -1/2 and (0 ≤ x ≤ π).

0 ≤ x ≤ π

So the right triangle should be in

either quadrant I or quadrant II.

See cos x = (-1/2).

Cosine is CAH:

Cosine,

Adjacent side (-1),

Hypotenuse (2).

So draw a right triangle on a coordinate plane

whose adjacent side is -1

and whose hypotenuse is 2.

Draw the angle x

that starts from the 3 o'clock position.

Find the missing side

by using the Pythagorean theorem:

(-1)^{2} + [height]^{2} = 2^{2}.

Then the height is √3.

This is a right triangle

whose sides are -1, √3, and 2.

So this is a 30-60-90 triangle.

So the central angle is, 60º, π/3.

Radian Measure

π/3 and x are supplementary.

So x = π - π/3.

So 2π/3 is the answer.