# Reference Angle

How to find the reference angle of the given angle: formula, 3 examples, and their solutions.

## Formula

The reference angle is the smaller angle

formed by the x-axis and the terminal side.

### Quadrant I: 0 ≤ θ ≤ π/2

If θ is in quadrant I,

then the reference angle is

θ.

### Quadrant II: π/2 ≤ θ ≤ π

If θ is in quadrant II,

then the reference angle is

π - θ.

### Quadrant III: π ≤ θ ≤ 3π/2

If θ is in quadrant III,

then the reference angle is

θ - π.

### Quadrant IV: 3π/2 ≤ θ ≤ 2π

If θ is in quadrant IV,

then the reference angle is

2π - θ.

## Example 1

### Example

### Solution

2π/3 is between π/2 and π.

So draw the terminal side on quadrant II.

Draw the reference angle.

It's the smaller angle

formed by the x-axis and the terminal side.

Then the reference angle is

π - 2π/3.

π = 3π/3

3π/3 - 2π/3 = π/3

So π/3 is the answer.

## Example 2

### Example

### Solution

5π/4 is between π and 3π/2.

So draw the terminal side on quadrant III.

Draw the reference angle.

It's the smaller angle

formed by the x-axis and the terminal side.

Then the reference angle is

5π/4 - π.

-π = -4π/4

5π/4 - 4π/4 = π/4

So π/4 is the answer.

## Example 3

### Example

### Solution

11π/6 is between 3π/2 and 2π.

So draw the terminal side on quadrant IV.

Draw the reference angle.

It's the smaller angle

formed by the x-axis and the terminal side.

Then the reference angle is

2π - 11π/6.

2π = 12π/6

12π/6 - 11π/6 = π/6

So π/6 is the answer.