How to Find Reference Angles?

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}})

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