Last modified on April 22nd, 2021

chapter outline

 

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.

Coterminal Angles

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°

Last modified on April 22nd, 2021

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