Find coterminal angles A_{c} to a given angle A.
What are coterminal angles?
If you graph angles x = 30^{o} and y =  330^{o} 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 Pi). Hence
A_{c} = A + k*360^{o} if A is given in degrees.
or
A_{c} = A + k*(2 PI) 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^{o}
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^{o}, 2(360)^{o} = 720^{o} (or any other positive angle multiple of 360^{o}). A positive coterminal angle A_{c} may be given by
A_{c} = 200^{o} + 360^{o} = 160^{o}
A negative coterminal angle to angle A may be obtained by adding 360^{o}, 2(360)^{o} = 720^{o} (or any other negative angle multiple of 360^{o}). A negative coterminal angle A_{c} may be given by
A_{c} = 200^{o}  360^{o} = 560^{o}
Example 2: Find a coterminal angle A_{c} to angle A =  17 Pi / 3 such that A_{c} is greater than or equal to 0 and smaller than 2 Pi
Solution to example 2:
A positive coterminal angle to angle A may be obtained by adding 2 Pi, 2(2 Pi) = 4 Pi (or any other positive angle multiple of 2 Pi). A positive coterminal angle A_{c} may be given by
A_{c} =  17 Pi / 3 + 2 Pi = 11 Pi / 3
As you can see adding 2*Pi 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 Pi)  x, where n is positive integer and x is a positive angle such that x < 2 pi.
 17 Pi /3 =  12 Pi / 3  5 Pi / 3 =  2 (2 Pi)  5 Pi / 3
From the above we can deduce that to make our angle positive, we need to add 3(2*Pi) = 6 Pi
A_{c} =  17 Pi /3 + 6 Pi = Pi / 3
Example 3: Find a coterminal angle A_{c} to angle A = 35 Pi / 4 such that A_{c} is greater than or equal to 0 and smaller than 2 Pi
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(2Pi) + x so that we can "see" what angle to add.
A = 35 Pi / 4 = 32 Pi / 4 + 3 Pi / 4 = 4(2 Pi) + 3 Pi /4
From the above we can deduce that to make our angle smaller than 2 Pi we need to add  4(2Pi) =  8 Pi to angle A
A_{c} = 35 Pi / 4  8 pi = 3 Pi /4
Exercises: (see solutions below)
1. Find a positive coterminal angle smaller than 360^{o} to angles
a) A = 700^{o} , b) B = 940^{o}
2. Find a positive coterminal angle smaller than 2 Pi to angles
a) A =  29 Pi / 6 , b) B = 47 Pi / 4
Solutions to Above Exercises:
1.
a) A_{c} = 20^{o} , b) B_{c} = 220^{o}
2. Find a positive coterminal angle smaller than 2 Pi to angles
a) A_{c} = 7 Pi / 6 , b) B_{c} = 7 Pi / 4
More references on angles.
