# How to Find Reference Angles?

The reference angle is the smallest angle you can make from the terminal side of an angle with the $$x$$-axis. In this step-by-step guide, you will learn more about reference angles.

## Step-by-step guide tofinding reference angles

The reference angle is the smallest possible angle formed by the terminal side of the given angle with the $$x$$-axis. It is always an acute angle (except when it is exactly $$90°$$). A reference angle is always positive regardless of which side the axis is falling.

To draw a reference angle for an angle, specify its end side and see at what angle the terminal side is closest to the $$x$$-axis.

### Rules for reference angles in each quadrant

Here are the reference angle formulas depending on the angle quadrant:

### Steps to find reference angles

The steps to find the reference angle of an angle are as follows:

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 forReference Angles

### Find the reference angle.

1. $$\color{blue}{\frac{31\pi }{9}}$$
2. $$\color{blue}{-250^{\circ }}$$
3. $$\color{blue}{-\frac{25\pi }{18}}$$
1. $$\color{blue}{\frac{4\pi }{9}}$$
2. $$\color{blue}{70^{\circ }}$$
3. $$\color{blue}{\frac{7\pi }{18}}$$

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