Use the Law of Tangents with SAS
How to Square Both Sides of a Trig Equation
How to Find the Area of a Triangle with SAS

How to Find Solutions for a Multiple-Angle Trigonometry Function

Multiple-angle trig functions include

image0.png

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

image1.png

in the equation

image2.png

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

image3.png

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

image4.png

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

image5.png

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

image6.png
  1. Write the inverse equation.

    image7.png
  2. List all the angles in two rotations,

    image8.png
  3. that have a sine with that value, and set them equal to

    image9.png

    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

    image10.png

    Notice how all the solutions

    image11.png

    are between 0 and 360 degrees — just as asked.

    image12.png
  5. Write the inverse equation.

    image13.png
  6. List all the angles in three rotations,

    image14.png

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

    image15.png

    The second two angles are just

    image16.png

    greater than the second two.

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

    image17.png

    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.

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