Find Coterminal Angles

Find coterminal angles A_c to a given angle A.

What are coterminal angles?

If you graph angles ? = 30° and ? = - 330° in standard position, these angles will have the same terminal side and are therefore called coterminal angles. See figure below.

Coterminal angles A_c to angle A may be obtained by adding or subtracting k × 360 degrees or k × (2 ?). Hence
A_c = A + k × 360° if A is given in degrees.
or
A_c = A + k ×(2 ?) if A is given in radians.
where k is any negative or positive integer.

Examples

Example 1

Find a positive and a negative coterminal angles to angle A = -200°

Solution to example 1:
There is an infinite number of possible answers to the above question since k in the formula for coterminal angles is any positive or negative integer.
A positive coterminal angle to angle A may be obtained by adding 360°, 2(360)° = 720° (or any other positive angle multiple of 360°). A positive coterminal angle A_c may be given by
A_c = -200° + 360° = 160°
A negative coterminal angle to angle A may be obtained by adding -360°, -2(360)° = -720° (or any other negative angle multiple of 360°). A negative coterminal angle A_c may be given by
A_c = -200° - 360° = -560°

Example 2

Find a coterminal angle A_c to angle A = - 17 ? / 3 such that A_c is greater than or equal to 0 and smaller than 2 ?

Solution to example 2:
A positive coterminal angle to angle A may be obtained by adding 2 ?, 2(2 ?) = 4 ? (or any other positive angle multiple of 2 ?). A positive coterminal angle A_c may be given by
A_c = - 17 ? / 3 + 2 ? = -11 ? / 3
As you can see adding 2*? is not enough to obtain a positive coterminal angle and we need to add a larger angle but what is the size of the angle to add?. We need to write our negative angle in the form - n (2 ?) - x, where n is positive integer and x is a positive angle such that x < 2 ?.
- 17 ? /3 = - 12 ? / 3 - 5 ? / 3 = - 2 (2 ?) - 5 ? / 3
From the above we can deduce that to make our angle positive, we need to add 3(2*?) = 6 ?
A_c = - 17 ? /3 + 6 ? = ? / 3

Example 3

Find a coterminal angle A_c to angle A = 35 ? / 4 such that A_c is greater than or equal to 0 and smaller than 2 ?

Solution to example 3:
We will use a similar method to that used in example 2 above: First rewrite angle A in the form n(2?) + x so that we can "see" what angle to add.
A = 35 ? / 4 = 32 ? / 4 + 3 ? / 4 = 4(2 ?) + 3 ? /4
From the above we can deduce that to make our angle smaller than 2 ? we need to add - 4(2?) = - 8 ? to angle A
A_c = 35 ? / 4 - 8 ? = 3 ? /4

Exercises

(see solutions below)
1. Find a positive coterminal angle smaller than 360° to angles
a) A = -700° , b) B = 940°
2. Find a positive coterminal angle smaller than 2 ? to angles
a) A = - 29 ? / 6 , b) B = 47 ? / 4

Solutions to Above Exercises:
1.
a) A_c = 20° , b) B_c = 220°
2. Find a positive coterminal angle smaller than 2 ? to angles
a) A_c = 7 ? / 6 , b) B_c = 7 ? / 4