# Coterminal Angle

How to find the coterminal angle of the given angle: definition, formula, 5 examples, and their solutions.

## Definition

### Definition

The angle on a coordinate is formed by

the x-axis and the terminal side.

So, coterminal angles are the angles

that have the same terminal side.

This angle θ and below angles

are coterminal angles

because they have the same terminal side.

## Formula

### Formula

As you can see,

coterminal angles have 360⋅n part.

(360⋅n means the number of rotation counterclockwise.)

So, to find the coterminal angles,

add 360⋅n (degree) or 2π⋅n (radian)

to the given angle θ.

(n is an integer.)

360⋅n: 360, 720, 1080, ...

2π⋅n: 2π, 4π, 6π, ...

## Example 1

### Example

### Solution

To find the coterminal angles,

add 360⋅n.

(Choose any integer n.)

Put 1 into the n.

Then the first coterminal angle of 60º is

360⋅1 + 60 degrees.

360 + 60 = 420

So 420º is the first coterminal angle.

Put 2 into the n.

Then the next coterminal angle is

360⋅2 + 60 degrees.

360⋅2 = 720

720 + 60 = 780

So 780º is the second coterminal angle.

Put -1 into the n.

Then the third coterminal angle is

360⋅(-1) + 60 degrees.

-360 + 60 = -300

So -300º is the third coterminal angle.

So 420º, 780º, and -300º

are the coterminal angles.

So 420º, 780º, -300º is the answer.

## Example 2

### Example

### Solution

To find the coterminal angles,

add 2π⋅n.

(Choose any integer n.)

Put 1 into the n.

Then the first coterminal angle of π/4 is

2π⋅1 + π/4.

Then 9π/4.

So 9π/4 is the first coterminal angle.

Put 2 into the n.

Then the next coterminal angle of π/4 is

2π⋅2 + π/4.

Then 17π/4.

So 17π/4 is the second coterminal angle.

Put 3 into the n.

Then the third coterminal angle of π/4 is

2π⋅3 + π/4.

Then 25π/4.

So 25π/4 is the third coterminal angle.

So 9π/4, 17π/4, and 25π/4

are the coterminal angles.

So 9π/4, 17π/4, 25π/4 is the answer.

## Example 3

### Example

### Solution

Write the 360⋅n numbers

that seems to cover 1000

on a number line:

0, 360, 720, 1080.

1000 is between 720 and 1080.

Then the number on the left side of 1000, 720,

is the 360⋅n number.

And the number between 720 and 1000

is the coterminal angle θ.

This means

720 + θ = 1000.

θ = [right angle] - [left angle]

So θ = 1000 - 720.

1000 - 720 = 280

So 280º is the answer.

## Example 4

### Example

### Solution

Write the 360⋅n numbers

that seems to cover -520

on a number line:

0, -360, -720, -1080.

-520 is between -720 and -360.

Then the number on the left side of -520, -720,

is the 360⋅n number.

And the number between -720 and -520

is the coterminal angle θ.

This means

-720 + θ = -520.

θ = [right angle] - [left angle]

So θ = -520 - (-720).

-520 - (-720)

= -520 + 720

= 200

So 200º is the answer.

## Example 5

### Example

### Solution

Write the 2π⋅n numbers

that seems to cover 13π/2

on a number line:

2π, 4π, 6π, 8π.

The denominator of 13π/2 is 2.

So change the denominators of the 2π⋅n numbers

to 2:

2π = 4π/2

4π = 8π/2

6π = 12π/2

8π = 16π/2.

13π/2 is between 12π/2 and 16π/2.

Then the number on the left side of 13π/2, 12π/2,

is the 2π⋅n number.

And the number between 12π/2 and 13π/2

is the coterminal angle θ.

This means

12π/2 + θ = 13π/2.

θ = [right angle] - [left angle]

So θ = 13π/2 - 12π/2.

13π/2 - 12π/2 = π/2

So π/2 is the answer.