Find Coterminal Angles

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 ansers 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

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