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.