# Coterminal Angles

See how to find the coterminal angles of the given angle.

5 examples and their solutions.

## Coterminal Angles

### Definition

θ(360 + θ)°(360⋅2 + θ)°

that have the same terminal side (↗).

### Formula

360n + θ (°)

2nπ + θ (rad)

To find the coterminal angles,2nπ + θ (rad)

add or subtract the multiples of 360° (2π).

360, 720, 1080, ...

2π, 4π, 6π, ...

Radian

### Example

Three coterminal angles of 60°

Solution 60 + 360 = 420

60 + 720 = 780- [1]

60 - 360 = -300

420°, 780°, -300°

60 + 720 = 780- [1]

60 - 360 = -300

420°, 780°, -300°

[1]

You can also find this by 420 + 360 = 780.

Close

### Example

Three coterminal angles of π4

Solution π4 + 2π

= π4 + 8π4

= 9π4

π4 + 4π - [1]

= π4 + 16π4

= 17π4

π4 + 6π - [2]

= π4 + 24π4

= 25π4

9π4, 17π4, 25π4

= π4 + 8π4

= 9π4

π4 + 4π - [1]

= π4 + 16π4

= 17π4

π4 + 6π - [2]

= π4 + 24π4

= 25π4

9π4, 17π4, 25π4

[1]

You can also find this by 9π/4 + 2π.

[2]

You can also find this by 17π/4 + 2π.

Close

### Example

θ: coterminal angle of 1000°

θ = ?

(0 ≤ θ ≤ 360°)

Solution θ = ?

(0 ≤ θ ≤ 360°)

θ = 1000 - 720- [1]

= 280°- [2] [3]

= 280°- [2] [3]

[1]

Add or subtract the multiples of 360

to make 0 ≤ θ ≤ 360°.

to make 0 ≤ θ ≤ 360°.

[2]

If the result is not in 0 ≤ θ ≤ 360°,

add or subtract the multiples of 360 again.

add or subtract the multiples of 360 again.

[3]

Don't forget to write the unit °.

Close

### Example

θ: coterminal angle of -1000°

θ = ?

(0 ≤ θ ≤ 360°)

Solution θ = ?

(0 ≤ θ ≤ 360°)

θ = -1000 + 1080

= 80°

= 80°

Close

### Example

θ: coterminal angle of 13π2

θ = ?

(0 ≤ θ ≤ 2π)

Solution θ = ?

(0 ≤ θ ≤ 2π)

θ = 13π2 - 6π - [1]

= 13π2 - 12π2 - [2]

= π2

= 13π2 - 12π2 - [2]

= π2

[1]

Add or subtract the multiples of 2π

to make 0 ≤ θ ≤ 2π.

to make 0 ≤ θ ≤ 2π.

[2]

If the result is not in 0 ≤ θ ≤ 2π,

add or subtract the multiples of 2π again.

add or subtract the multiples of 2π again.

Close