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.