How to Find Solutions for a Multiple-Angle Trigonometry Function

Multiple-angle trig functions include

and so on. When considering inverse relations (which give multiple answers) for these angles, the multiplier helps you determine the number of answers to expect. You take the number of answers you find in one full rotation and take that times the multiplier. For example, if you’re looking for

in the equation

then you get two different answers if you consider all the angles between 0 and 360 degrees:

equals 60 and 120 degrees. But if you change the equation to

you get twice as many, or four, answers between 0 and 360 degrees:

equals 30, 60, 210, and 240 degrees. These angles are all within one rotation, but putting them into the original equation and multiplying gives angles with the same terminal side as the angles within one rotation.

Here are some examples to show you how this multiplication works and how to find the answers. First, to show you how to get the answers to

1. Write the inverse equation.

   

2. List all the angles in two rotations,

   

3. that have a sine with that value, and set them equal to

   

   The second two angles are just 360 more than the corresponding first two.

4. Divide the terms on both sides of the equation by 2 to solve for

   

   Notice how all the solutions

   

   are between 0 and 360 degrees — just as asked.

   

5. Write the inverse equation.

   

6. List all the angles in three rotations,

   

   that have that cosine, and set them equal to 3x.

   

   The second two angles are just

   

   greater than the second two.

7. Multiply both sides by 1/3 to solve for x.

   

   This result shows the big advantage of radians — the numbers don’t get as big as they do with degrees. The disadvantage may be having so many fractions.