Coterminal Angles

Coterminal Angles

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

Formula

(coterminal angle) = 360n + (degree) = 2*pi*n + (radian)

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)
2πn + 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, 2π⋅1 + θ, and 2π⋅2 + θ (in radian).

Example 1

For each given angle, find a coterminal angle theta. (0 <= theta <= 360) 1. 1000 degree, 2. -60 degree

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

For each given angle, find a coterminal angle theta. (0 <= theta <= 2pi) 1. 13*pi/2, 2. -2*pi/3

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

Then 13 = 4⋅3 + 1.

So 13π/2 = 2π⋅3 + 1⋅π/2

So π/2 is the coterminal angle.

13π/2 = 2π⋅3 + π/2 means
the angle rotates 3 times counterclockwise (2π⋅3)
and goes π/2 counterclockwise.

To make a coterminal angle between 0 and 2π,
write -2π and +2π.

Then -2π/3 = 2π⋅(-1) + 4π/3.

So the coterminal angle is 4π/3.

-2π/3 = 2π⋅(-1) + 4π/3 means
the angle rotates 1 time clockwise [2π⋅(-1)]
and goes 4π/3 counterclockwise.