How to Find Coterminal Angles (Definition and Application)

Do you want to know how to find coterminal angles? Angles that begin and terminate at the same location are said to be coterminal angles. They have a common beginning and end. Notably, they need not be at the same angle.

How to find conterminal angles

How to Find Coterminal Angles

The angles that share a terminal side and an initial side are said to be coterminal.

To calculate the coterminal angles:

▸If the angle is in degrees, multiply or divide by multiples of 360.

▸If the angle is in radians, add or subtract multiples of 2 from the provided angle.

Therefore, in order to get the coterminal angles, we really do not need to apply the coterminal angles formula.

Instead, to get the coterminal angles of an angle, we can either add or remove multiples of 360° (or 2).

Let’s use the provided example to assist us to understand the topic.

Find the coterminal angle of π/4., for instance.


The specified angle, which is expressed in radians, is = π/4.

Therefore, we multiply it by multiples of 2 to determine its coterminal angles.

Let’s take 2 away from the indicated angle.

π/4 − 2π = −7π/4

Thus, a coterminal angle of π/4 is −7π/4. is therefore 7/4.

When you fully grasp the idea, you will distinguish between coterminal angles and reference angles and use the coterminal angles formula to solve issues.

What Are Coterminal Angles?

The word “coterminal” refers to angles that have the same starting side and terminal side.

Although these angles have different values, they are situated where they usually are.

Both have the same sides and quadrant and the same vertices. The terminal sides are at the same angle regardless of whether the angles are turned in a clockwise or counterclockwise direction.

The angle represents the rotation of a beam regarding its origin.

The angle’s beginning side represents the location of the beam’s initial start, while its terminal side represents the location of the ray after it has rotated.


Coterminal Angles Formula

Whether an angle is expressed in terms of degrees or radians, the following formula can help you determine its coterminal angles:

▸Degrees: θ ± 360 n

▸Radians: θ ± 2πn

In the formula above, n is an integer that shows the number of rotations around the coordinate plane, and the term “360n” refers to a multiple of 360.

The following angles are all coterminal angles: 45°, -315°, 405°, – 675°, 765°, etc. They only differ by a certain number of whole circles.

According to this definition, “two angles are said to be coterminal if the difference between the angles is a multiple of 360° (or 2, if the angle is in terms of radians)”.

Let us study the concept with the aid of the example.

Example: Find two coterminal angles of 30°.


The angle is, θ = 30°

The formula is, θ ± 360n

Find two coterminal angles together.

n = 1 for determining a single coterminal angle (anticlockwise). The coterminal angle that corresponds is thus,

= θ + 360n

= 30 + 360 (1)

= 390°

Calculating a second coterminal angle: n = 2 (clockwise)

The coterminal angle that corresponds is thus,

= θ + 360n

= 30 + 360(−2)

= −690°

Positive and Negative Coterminal Angles

There are two possible coterminal angles: positive and negative. We discovered that 390° and -690° are the coterminal angles of 30° in one of the aforementioned situations.

▸The positive coterminal angle of 30° is 390°.

▸The coterminal angle of 30° that is negative is -690°.

θ ± 360 n, where n changes from a negative number when the rotation is clockwise to a positive value when it is anticlockwise.

We choose whether to add or remove multiples of 360° (or 2π) to get positive or negative coterminal angles.

Coterminal Angles and Reference Angles

The coterminal angles of a given angle can already be determined.

The reference angle for every angle is the angle between the angle’s terminal side and the x-axis, and it is always between 0° and 90°.

The terminal side of the quadrant determines the reference angle.

According to the terminal side’s quadrant, we must follow the following procedures to get the reference angle of an angle:

▸We start by calculating its coterminal angle, which ranges from 0° to 360°.

▸The coterminal angle’s quadrant is then visible.

▸The reference angle is the same as our specified angle if the terminal side is in the first quadrant (0° to 90°). For instance, if the angle is 25 degrees, then the reference angle is likewise 25 degrees.

▸The reference angle is equal to 180 degrees minus the given angle if the terminal side is in the second quadrant (90 to 180 degrees). If the angle is 100°, for instance, the reference angle is 180° – 100° = 80°.

▸The reference angle is (given angle – 180°) if the terminal side is in the third quadrant (180° to 270°).

For instance, if an angle is provided at 215°, its reference angle is 215° – 180°, which equals 35°.

Depending on whether the given angle is in degrees or radians, finding coterminal angles is as easy as arithmetically adding or subtracting 360° or 2π from each angle.

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CSN Team.

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