What are coterminal angles?

If you graph angles x = 30° and y = - 330° in standard position, these angles will have the same terminal side. See figure below.

Coterminal angles A_{c} to angle A may be obtained by adding or subtracting k*360 degrees or k* (2 π). Hence

A_{c} = A + k*360° if A is given in degrees.

or

A_{c} = A + k*(2 π) if A is given in radians.

where k is any negative or positive integer.

__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 π / 3 such that A_{c} is greater than or equal to 0 and smaller than 2 π

__Solution to example 2:__

A positive coterminal angle to angle A may be obtained by adding 2 π, 2(2 π) = 4 π (or any other positive angle multiple of 2 π). A positive coterminal angle A_{c} may be given by

A_{c} = - 17 π / 3 + 2 π = -11 π / 3

As you can see adding 2*π 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 π) - x, where n is positive integer and x is a positive angle such that x < 2 π.

- 17 π /3 = - 12 π / 3 - 5 π / 3 = - 2 (2 π) - 5 π / 3

From the above we can deduce that to make our angle positive, we need to add 3(2*π) = 6 π

A_{c} = - 17 π /3 + 6 π = π / 3

__Example 3:__ Find a coterminal angle A_{c} to angle A = 35 π / 4 such that A_{c} is greater than or equal to 0 and smaller than 2 π

__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(2π) + x so that we can "see" what angle to add.

A = 35 π / 4 = 32 π / 4 + 3 π / 4 = 4(2 π) + 3 π /4

From the above we can deduce that to make our angle smaller than 2 π we need to add - 4(2π) = - 8 π to angle A

A_{c} = 35 π / 4 - 8 π = 3 π /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 π to angles

a) A = - 29 π / 6 , b) B = 47 π / 4

__Solutions to Above Exercises:__

1.

a) A_{c} = 20° , b) B_{c} = 220°

2. Find a positive coterminal angle smaller than 2 π to angles

a) A_{c} = 7 π / 6 , b) B_{c} = 7 π / 4

More references on angles.