 # Find Coterminal Angles

 Find coterminal angles Ac 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 Ac to angle A may be obtained by adding or subtracting k*360 degrees or k* (2 π). Hence Ac = A + k*360� if A is given in degrees. or Ac = 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 Ac may be given by Ac = -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 Ac may be given by Ac = -200� - 360� = -560� Example 2: Find a coterminal angle Ac to angle A = - 17 π / 3 such that Ac 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 Ac may be given by Ac = - 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 Ac = - 17 π /3 + 6 π = π / 3 Example 3: Find a coterminal angle Ac to angle A = 35 π / 4 such that Ac 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 Ac = 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) Ac = 20� , b) Bc = 220� 2. Find a positive coterminal angle smaller than 2 π to angles a) Ac = 7 π / 6 , b) Bc = 7 π / 4 More references on angles.