Find coterminal angles A_{c} to a given angle A.
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.
