Coterminal Angles

What are Coterminal Angles

Two or more angles are called coterminal angles if they are in standard position having their initial side on the positive x-axis and a common terminal side. Based on the direction of rotation, coterminal angles can be positive or negative.

In the above figure, 45°, 405° and -315° are coterminal angles having the same initial side (x-axis) and the same terminal side but with different amount of rotations.

Other Examples: Similarly, 30°, -330°, 390° and 57°, 417°, -303° are also coterminal angles.

Formula

How to Find Coterminal Angles

We can find the coterminal angles of a given angle by either adding or subtracting a multiple of 360°,if the angle is measured in degree or 2π, if the angle is measured in radians.

For example, the coterminal angles of a given angle θ can be obtained using the given formula:

i) For positive coterminal angles = θ + 360° x k, if θ is given in degrees, and k is an integer

ii) For positive coterminal angles = θ + 2π x k, if θ is given in radians, and k is an integer

iii) For negative coterminal angles = θ – 360° x k, if θ is given in degrees, and k is an integer

iv) For negative coterminal angles = θ – 360° x k, if θ is given in radians, and k is an integer

Thus two angles are coterminal if the differences between them are a multiple of 360° or 2π

Find a positive and a negative coterminal angle of 35°.

Solution:

One positive coterminal angle with 35° is:
35° + 360° = 395°
One negative coterminal angle with 35° is:
35° – 360° = -325°

Find a positive and a negative coterminal angle of π/2.

Solution:

As we know,
Positive coterminal angles of π/2 in radian = π/2 + 2π
= 5 π/2
Similarly,
Negative coterminal angles of π/2 in radian = π/2 – 2π
= -3 π/2

Determine if the flowing pairs of angles are coterminal.

Solution:

(a) -520°, 200°
(b) 417°, -303°
(c) –600°, –40°
As we know,
The measurements of coterminal angles differ by an integer multiple of 360°
(a)  –520° – 200° = –720° = –2(360°), which is a multiple of 360°
Hence, –520 and 200° are coterminal angles
(b) 417° − (−303°) =  720°  =  2(360°), which is a multiple of 360°
Hence, 417°and -303° are coterminal angles
(c) –600° – (–40°) = –560°, which is not a multiple of 360°
Hence, –600° and –40° are not coterminal angles

Find an angle between -500 and +500 and that is coterminal with θ = 75°.

Solution:

As we know,
The measurements of coterminal angles differ by an integer multiple of 360°
For θ between 500° and 0°, the coterminal angles are 75° and 75° + 360°= 435°
For θ between 0° and – 500°, the coterminal angle is 75° – 360° = -285°