How to Find Reference Angles?

How to find reference angles, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.

Rules for reference angles in each quadrant

Steps to find reference angles

1. Find the coterminal angle of the given angle that lies between (0°) and (360°).
2. If the angle of step (1) is between (0) and (90°), that angle itself is the reference angle of the given angle. If not, then we need to check if it is close to (180°) or (360°) and how much.
3. The angle from step (2) is the angle reference angle.

Reference Angles , Example 1:

Find the reference angle of (frac{8π}{3}) in radians.

Solution:

First, find the coterminal angle. To find its coterminal angle subtract (2π) from it.

(frac{8π}{3} – 2π = frac{2π}{3})

This angle is not between (0) and (frac{π}{2}). Therefore, it is not the reference angle of the given angle. Then check whether (frac{2π}{3}) is close to (π) or (2π) and by how much.

(frac{2π}{3}) is close to (π) by (π – frac{2π}{3} = frac{π}{3}). Therefore, the reference angle of (frac{8π}{3}) is (frac{π}{3}).

Exercises for Reference Angles

Find the reference angle.

1. (color{blue}{frac{31pi }{9}})
2. (color{blue}{-250^{circ }})
3. (color{blue}{-frac{25pi }{18}})

1. (color{blue}{frac{4pi }{9}})
2. (color{blue}{70^{circ }})
3. (color{blue}{frac{7pi }{18}})

The Ultimate Algebra Bundle 2026 From Pre-Algebra to Algebra II

Download

$109.99 Original price was: $109.99.$54.99Current price is: $54.99.

The Ultimate Elementary School Math Bundle Grade 3-5

Download

$109.99 Original price was: $109.99.$54.99Current price is: $54.99.