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.