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.