Arccosine: Value

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).

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 = 2².

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.

π - π/3
= 3π/3 - π/3
= 2π/3

Quotient Identities

So 2π/3 is the answer.