# Coterminal Angles

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

## Formula

Coterminal angles are angles

whose terminal ray (white ray)

are in the same position.

These three angles' terminal rays

are in the same position.

So these three angles are coterminal angles.

The measures of coterminal angles are:

360⋅*n* + *θ* (degree)

2*π*⋅*n* + *θ* (radian)*n* is the number of counterclock rotation.

So the measures of above three angles are*θ*, 360⋅1 + *θ*, and 360⋅2 + *θ* (in degree)*θ*, 2*π*⋅1 + *θ*, and 2*π*⋅2 + *θ* (in radian).

## Example 1

Divide 1000 by 360.

Then 1000 = 360⋅2 + 280.

So, the remainder, 280º is the coterminal angle

that is between 0º and 360º.

1000 = 360⋅2 + 280 means

the angle rotates 2 times counterclockwise (360⋅2)

and goes 280º counterclockwise.

To make a coterminal angle between 0º and 360º,

write -360 and +360.

Then -60 = -360⋅(-1) + 300.

So the coterminal angle is 300º.

-60 = 360⋅(-1) + 300 means

the angle rotates 1 time clockwise [360⋅(-1)]

and goes 300º counterclockwise.

## Example 2

Divide 13 by 4. [4 = 2⋅(2, denominator)]

Then 13 = 4⋅3 + 1.

So 13*π*/2 = 2*π*⋅3 + 1⋅*π*/2

So *π*/2 is the coterminal angle.

13*π*/2 = 2*π*⋅3 + *π*/2 means

the angle rotates 3 times counterclockwise (2*π*⋅3)

and goes *π*/2 counterclockwise.

To make a coterminal angle between 0 and 2*π*,

write -2*π* and +2*π*.

Then -2*π*/3 = 2*π*⋅(-1) + 4*π*/3.

So the coterminal angle is 4*π*/3.

-2*π*/3 = 2*π*⋅(-1) + 4*π*/3 means

the angle rotates 1 time clockwise [2*π*⋅(-1)]

and goes 4*π*/3 counterclockwise.