# Cosine: Value

How to find the value of cosine (trigonometry): 1 example and its solution. + Cosine values of commonly used angles.

## Example

### Example

### Solution

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.

## Cosine Values of Commonly Used Angles

### Table

These are the cosine values

of commonly used angles.

### Detail

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.