Coterminal Angles

The coterminal angles are the angles that have the same initial side and the same terminal sides. We determine the coterminal angle of a given angle by adding or subtracting 360° or 2π to it. In trigonometry, the coterminal angles have the same values for the functions of sin, cos, and tan.

Once you have understood the concept, you will differentiate between coterminal angles and reference angles, as well as be able to solve problems with the coterminal angles formula.

Coterminal angles are the angles that have the same initial side and share the terminal sides. These angles occupy the standard position, though their values are different. They are on the same sides, in the same quadrant and their vertices are identical. When the angles are moved clockwise or anticlockwise the terminal sides coincide at the same angle. An angle is a measure of the rotation of a ray about its initial point. The original ray is called the initial side and the final position of the ray after its rotation is called the terminal side of that angle.

Consider 45°. Its standard position is in the first quadrant because its terminal side is also present in the first quadrant. Look at the image.

• On full rotation anticlockwise, 45° reaches its terminal side again at 405°. 405° coincides with 45° in the first quadrant.
• On full rotation clockwise, 45° reaches its terminal side again at -315°. -315° coincides with 45° in the first quadrant.

Thus 405° and -315° are coterminal angles of 45°.

The formula to find the coterminal angles of an angle θ depending upon whether it is in terms of degrees or radians is:

• Degrees: θ ± 360 n
• Radians: θ ± 2πn

In the above formula, θ ± 360n, 360n means a multiple of 360, where n is an integer and it denotes the number of rotations around the coordinate plane.

Thus we can conclude that 45°, -315°, 405°, - 675°, 765° ..... are all coterminal angles. They differ only by a number of complete circles. We can conclude that "two angles are said to be coterminal if the difference between the angles is a multiple of 360° (or 2π, if the angle is in terms of radians)". Let us learn the concept with the help of the given example.

Example : Find two coterminal angles of 30°.

Solution:

The given angle is, θ = 30°

The formula to find the coterminal angles is, θ ± 360n

Let us find two coterminal angles.

For finding one coterminal angle: n = 1 (anticlockwise)

Then the corresponding coterminal angle is,

= θ + 360n
= 30 + 360 (1)
= 390°

Finding another coterminal angle :n = −2 (clockwise)

Then the corresponding coterminal angle is,

= θ + 360n
= 30 + 360(−2)
= −690°

From the above explanation, for finding the coterminal angles:

• add or subtract multiples of 360° from the given angle if the angle is in degrees.
• add or subtract multiples of 2π from the given angle if the angle is in radians.

So we actually do not need to use the coterminal angles formula to find the coterminal angles. Instead, we can either add or subtract multiples of 360° (or 2π) from the given angle to find its coterminal angles. Let us understand the concept with the help of the given example.

Example: Find a coterminal angle of π/4.

Solution:

The given angle is θ = π/4, which is in radians.

So we add or subtract multiples of 2π from it to find its coterminal angles.

Let us subtract 2π from the given angle.

π/4 − 2π = −7π/4

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

The coterminal angles can be positive or negative. In one of the above examples, we found that 390° and -690° are the coterminal angles of 30°

Here,

• 390° is the positive coterminal angle of 30° and
• -690° is the negative coterminal angle of 30°

θ ± 360 n, where n takes a positive value when the rotation is anticlockwise and takes a negative value when the rotation is clockwise. So we decide whether to add or subtract multiples of 360° (or 2π) to get positive or negative coterminal angles.

We already know how to find the coterminal angles of a given angle. The reference angle of any angle always lies between 0° and 90°, It is the angle between the terminal side of the angle and the x-axis. The reference angle depends on the quadrant's terminal side.

The steps to find the reference angle of an angle depends on the quadrant of the terminal side:

• We first determine its coterminal angle which lies between 0° and 360°.
• We then see the quadrant of the coterminal angle.
• If the terminal side is in the first quadrant ( 0° to 90°), then the reference angle is the same as our given angle. For example, if the given angle is 25°, then its reference angle is also 25°.
• If the terminal side is in the second quadrant ( 90° to 180°), then the reference angle is (180° - given angle). For example, if the given angle is 100°, then its reference angle is 180° – 100° = 80°.
• If the terminal side is in the third quadrant (180° to 270°), then the reference angle is (given angle - 180°). For example, if the given angle is 215°, then its reference angle is 215° – 180° = 35°.
• If the terminal side is in the fourth quadrant (270° to 360°), then the reference angle is (360° - given angle). For example, if the given angle is 330°, then its reference angle is 360° – 330° = 30°.

Example: Find the reference angle of 495°.

Solution:

Let us find the coterminal angle of 495°. The coterminal angle is 495° − 360° = 135°.

The terminal side lies in the second quadrant. Thus the reference angle is 180° -135° = 45°

Therefore, the reference angle of 495° is 45°.

Important Notes on Coterminal Angles:

• The difference (in any order) of any two coterminal angles is a multiple of 360°
• To find the coterminal angle of an angle, we just add or subtract multiples of 360°. from the given angle.
• The number of coterminal angles of an angle is infinite because there is an infinite number of multiples of 360°.
• If two angles are coterminal, then their sines, cosines, and tangents are also equal.

☛Related Articles:

• Types of Angles and their Properties
• Pairs of Angles
• Corresponding Angles
• Coterminal Angles Calculator

What is the Process of Finding Coterminal Angles?

For finding coterminal angles, we add or subtract multiples of 360° or 2π from the given angle according to whether it is in degrees or radians respectively. For example, some coterminal angles of 10° can be 370°, -350°, 730°, -710°, etc. In other words, the difference between an angle and its coterminal angle is always a multiple of 360°.

What is the Coterminal Angle of 45°?

Let us find a coterminal angle of 45° by adding 360° to it. 45° + 360° = 405°. Thus, 405° is a coterminal angle of 45°.

What is the Coterminal Angle of 60°?

Let us find a coterminal angle of 60° by subtracting 360° from it. 60° − 360° = −300°. Thus, -300° is a coterminal angle of 60°.

What is the Coterminal Angle of -30° Between 0° and 360°?

To find a coterminal angle of -30°, we can add 360° to it. − 30° + 360° = 330°. Thus, 330° is the required coterminal angle of -30°.

How do you Find Positive Coterminal Angles?

To find positive coterminal angles we need to add multiples of 360° to a given angle. For example, the positive coterminal angle of 100° is 100° + 360° = 460°.

How do you Find Negative Coterminal Angles?

To find negative coterminal angles we need to subtract multiples of 360° from a given angle. For example, the negative coterminal angle of 100° is 100° - 360° = -260°.