Cosine: Value

First, find the reference angle of 4π/3.

4π/3 is between π and 3π/2.
So draw the terminal side on quadrant III.

Then the reference angle is
4π/3 - π.

-π = -3π/3

4π/3 - 3π/3 = π/3

So the reference angle is π/3.

Draw a right triangle like this.

The central angle is π/3: 60º.
So this is a 30-60-90 triangle.

So the base is -1.
The height is -√3.
And the hypotenuse is 2.

cos 4π/3 is the cosine of the right triangle.

Cosine is CAH:
Cosine,
Adjacent side (-1),
Hypotenuse (2).

So cos 4π/3 = -1/2.

Write the minus sign
in front of the fraction.

So cos 4π/3 = -1/2.

These are the cosine values
of commonly used angles.

θ = 0

The adjacent side (base) is 1.
The opposite side (height) is 0.
The hypotenuse is 1.

CAH:
Cosine,
Adjacent side (1),
Hypotenuse (1).

So cos 0 = 1/1 = 1.

θ = π/6

This is a 30-60-90 triangle.

The adjacent side (base) is √3.
The opposite side (height) is 1.
The hypotenuse is 2.

CAH:
Cosine,
Adjacent side (√3),
Hypotenuse (2).

So cos π/6 = √3/2.

θ = π/4

This is a 45-45-90 triangle.

The adjacent side (base) is 1.
The opposite side (height) is 1.
The hypotenuse is √2.

CAH:
Cosine,
Adjacent side (1),
Hypotenuse (√2).

So cos π/4 = 1/√2 (= √2/2).

θ = π/3

This is a 30-60-90 triangle.

The adjacent side (base) is 1.
The opposite side (height) is √3.
The hypotenuse is 2.

CAH:
Cosine,
Adjacent side (1),
Hypotenuse (2).

So cos π/3 = 1/2.

θ = π/2

The adjacent side (base) is 0.
The opposite side (height) is 1.
The hypotenuse is 1.

CAH:
Cosine,
Adjacent side (0),
Hypotenuse (1).

So cos π/2 = 0/1 = 0.