Coterminal angles are angles in coordinate plane that result out of rotation of the terminal side of angles in standard position. The angle in standard position has the vertex at the origin and one ray on the positive x -axis, which is called the initial side of the angle. The other ray is called the terminal side of the angle.

In the above example angle measure 45º has initial side on the x-axis and the terminal side along OP. If you imagine OP to be a circling ray, it comes back to the same position after completing one full rotation about the origin. Thus it has described an angle measure of 405º. Angles 45 and 405º both share the terminal ray along OP. If the rotation of OP is continued, it describes angles 765º, 1125º ..... after completing each rotation. All these angle measures are Coterminal angles.
The Coterminal angles makes it possible for the angle to have any real number as its measure and map all these measures to Unit circle angles.

Coterminal Angles Definition

Two angles in standard position are called coterminal angles if they have the same terminal side.
As coterminal angles result due to the rotation of the terminal side, they differ in multiples of 360º.
Coterminal angle to a given angle can be found by adding or subtracting multiples of 360 to the given angle.

An angle measure is positive if the rotation from the initial side to the terminal side is counter clockwise and negative if the rotation is clockwise. It is also possible for a positive angle and a negative angles to be coterminal angles.

The Quadrant angle 270º is formed by the counter clockwise rotation of the terminal side OP. When the terminal side is rotated in the clockwise direction, it coincides with OP when it describes 90º.
Thus the angles 270 and -90 are coterminal angles.
Thus Coterminal angles can also be defined as,
If θ is the measure of an angle then all angles with a measure of θ + 360n where n is an integer are Coterminal with θ.

Positive and Negative Coterminal Angles

While a positive angle is the measure of the rotation of its terminal side from positive x axis in anti clockwise direction, a negative angle is the measure of the rotation of the terminal side in the clockwise direction.
If a given angle is positive, a negative coterminal angle is got by subtracting 360 from it.

Example:
If θ = 45º, then θ' = 45 - 360 = -315 is a negative Coterminal Angle of 45º.
More negative Coterminal angles can be found by subtracting multiples of 360 from θ.
like 45 - 720 = -675º, 45 - 1080 = -1035º etc.

If a given angle is negative, then a positive coterminal angle is got by adding 360 to it.

Example:
If φ = -75º, then φ' = -75 + 360 = 285º is a positive Coterminal Angle of 75º.
Other positive coterminal angles can be found by adding multiples of 360 to φ
like -75 + 720 = 645º, -75 + 1080 = 1005º etc.

When θ is an angle given in radians, its Coterminal angles are got by adding even multiples of π.
Hence 360º = 2π.

Examples:
Let θ = $\frac{\pi }{3}$ radians.
Its is Coterminal angles are $\frac{\pi }{3}$ + 2π, $\frac{\pi }{3}$ + 4π..........and
$\frac{\pi }{3}$ - 2π, $\frac{\pi }{3}$ - 4π..........

The formula for finding coterminal angles when the angle is given in radian measure can be written as
θ + 2nπ where n is an integer.

How to find Coterminal Angles?

Let us summarize the method of finding Coterminal Angles in following steps:
1. Check the unit of the angle given degree or radian.
2. If the angle is given in degree measures use the formula θ + 360n where n is an integer. By giving positive integer values to n we get positive coterminal angles of the given angle. When n is given negative integer values, the negative coterminal angles are found.
3. If the angle is given in radian measures, use the formula θ + 2nπ where n is an integer. By giving positive integer values to n we get positive coterminal angles of the given angle. When n is given negative integer values, the negative coterminal angles are found.

Coterminal Angles Examples

Solved Examples

Question 1: Find five coterminal angles of 225º, two positive and three negative.
Solution:

The formula for finding the Coterminal angles is θ + 360n
As we need to find two positive coterminal angles, we need to apply the formula twice giving two positive integer values for n.
225º + 1 x 360º = 585º
225 + 2 x 360º  = 225º + 720º = 945º
To find three negative coterminal angles we need to give three negative values for n.
225 + (-1)360 = 225 - 360

Question 2: Which of the following angles are Coterminal angles?
(a)  330º
(b) 30º
(c) 390º
(d) -330º
Solution:

Angles 390º and -330º can be got by adding multiples of 360 to 30.
390 = 30 + 360    and -330 = 30 +(- 360) = 30 - 360
The terminal sides of angles 30º, 390º and -330º also fall in the first quadrant as shown in the sketch below.
330º cannot be expressed using the formula for finding the coterminal angle of any the other three angles given.
The terminal side of angle 330º falls in the fourth quadrant as seen in the sketch below.

Question 3: Determine a coterminal angle of 765º which is between 0º and 360º. Find also the quadrant in which the terminal side of 765º falls.
Solution:

As angle 765º is greater than 360º we need to subtract the multiples of 360 from 765 to find the coterminal angle in the
given range.
360 x 1 = 360
360 x 2 = 720 < 765
Subtracting 720 from 765, the coterminal angle in the given range = 765 -720 = 45º.

Question 4: Find one positive and one negative coterminal angles of $\frac{9\pi }{4}$ radians.
Solution:

To get the positive coterminal angle add 2π.
$\frac{9\pi }{4}$ + 2π = $\frac{9\pi }{4}$ + $\frac{8\pi }{4}$ = $\frac{17\pi }{4}$

The negative coterminal angle will be obtained by subtracting 4π as $\frac{9\pi }{4}$ > 2π.
$\frac{9\pi }{4}$ - 4π = $\frac{9\pi }{4}$ - $\frac{16\pi }{4}$ = $\frac{-7\pi }{4}$.