Coterminal angles are angles that have the same initial side and the same terminal side. The coterminal angles are equivalent to each other since they represent the same direction. It is possible to have both positive and negative coterminal angles.

Here, we will explore these concepts in more detail using diagrams and learn about the formula we can use to calculate coterminal angles in both degrees and radians. Then, we will apply these formulas to solve some practice problems.

## What are coterminal angles?

Coterminal angles are defined as angles that have the same initial side and the same terminal side. These angles are considered equivalent since they indicate the same direction. It is possible to have positive and negative terminal angles.

## What is the formula for coterminal angles?

The formula for finding the terminal angles of a given angle depends on whether the given angle is in radians or in degrees. Therefore, we have the following two formulas:

Degrees: $latex \theta\pm 360^{\circ} n$ Radians: $latex \theta\pm 2\pi n$ |

where *n *is any integer number.

In the example above, we have that 45° and -315° are coterminal. We can verify this by subtracting them and observing that their difference is 360°:

$latex 45^{\circ}-(-315^{\circ})=360^{\circ}$

We can also find another coterminal angle for the 45° angle:

In this case, we observe that the difference between both angles is -360°, which is a multiple of 360°:

$latex 45^{\circ}-405^{\circ}=-360^{\circ}$

We can conclude that two angles are considered coterminal if their difference is equal to a multiple of 360° or a multiple of 2π if the angle is in radians. Similarly, we can find coterminal angles of a given angle by adding a multiple of 360° or a multiple of 2π to the original angle.

## Positive and negative coterminal angles

In the example above, we find that 405° and -315° are the coterminal angles of 45°. Therefore, we have:

- 405° is the positive coterminal angle of 45°
- -315° is the negative coterminal angle of 45°

Then, we can decide if we want to add or subtract multiples of 360° or of 2π depending on whether we want to obtain a positive or negative coterminal angle.

## Coterminal angles – Examples with answers

The coterminal angle formulas are applied to solve the following examples. Each example has its respective solution, where you can look at the process and reasoning used.

**EXAMPLE 1**

Find two coterminal angles of 30°.

##### Solution

The given angle is $latex \theta=30^{\circ}$. The formula for finding the coterminal angles is:

$latex \theta \pm 360n$

*n* can be any integer number. We can use $latex n = 1$ and $latex n = -1$ to find two different coterminal angles. Therefore, we have:

$latex \theta \pm 360n$

$latex =30^{\circ} + 360(1)$

$latex =390^{\circ}$

Now, using $latex n=-1$, we have:

$latex \theta \pm 360n$

$latex =30^{\circ} + 360(-1)$

$latex =-330^{\circ}$

**EXAMPLE 2**

Find two coterminal angles of $latex \frac{\pi}{4}$.

##### Solution

We have the angle $latex \theta=\frac{\pi}{4}$. This angle is in radians, so we use the following formula to find the coterminal angles:

$latex \theta \pm 2\pi n$

*n* can be any integer number. In this case, we are going to use $latex n=2$ and $latex n=-1$. Therefore, we have:

$latex \theta \pm 2\pi n$

$latex =\frac{\pi}{4} + 2\pi(2)$

$latex =\frac{17\pi}{4}$

Now, using $latex n=-1$, we have:

$latex \theta \pm 2\pi n$

$latex =\frac{\pi}{4} + 2\pi(-1)$

$latex =-\frac{7\pi}{4}$

**EXAMPLE 3**

Find a positive coterminal angle for the angle -1500°.

##### Solution

We can perform the following division:

$latex \frac{1500}{360}=4.17$

This means that we have 4 full 360° angles in the -1500° angle. Therefore, we have to add 5 times 360° to get a positive angle. Using the coterminal angle formula with $latex n=5$:

$latex \theta \pm 360n$

$latex -1500^{\circ} + 360(5)$

$latex =300^{\circ}$

**EXAMPLE 4**

What is a positive coterminal angle for the angle $latex – \frac{10 \pi}{3}$?

##### Solution

Similar to the previous example, we carry out the following division:

$latex \frac{\frac{10\pi}{3}}{2\pi}=1.67$

We see that we have 1 complete angle of 2π, so we have to add at least 2 times 2π to obtain a positive angle. Using the coterminal angle formula in radians with $latex n = 2$, we have:

$latex \theta \pm 2\pi$

$latex -\frac{10\pi}{3} + 2\pi(2)$

$latex =\frac{2\pi}{3}$

## Coterminal angles – Practice problems

Apply the formulas for coterminal angles to solve the following problems. Select an answer and check it to see if you got the correct answer.

## See also

Interested in learning more about trigonometric identities? Take a look at these pages:

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