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Solve a Trig Equation by Finding a Greatest Common Factor

The trigonometry equations that require finding a greatest common factor, factoring it out, and then solving the equation could look like one of the following two equations:

image0.png

Both of these equations are solved here.

image1.png
  1. Factor out sin x from each of the two terms.

    image2.png
  2. Set the two different factors equal to 0.

    image3.png
  3. Solve for the values of x that satisfy both equations. Use a scientific calculator.

    image4.png

    All these values are solutions for the original equation. The complete list is x = 0°, 60°, 180°, 300°.

    image5.png
  4. Move the term on the right to the left by subtracting it from each side.

    image6.png

A common error in algebra — which could carry over in trig — is to divide each side by the common factor. You should never do that, because you’ll lose part of the solution.

  1. Factor out the cos x from each term.

    image7.png

    You don’t want to divide each side by cos x, because you’ll lose a solution if you do.

  2. Set the two different factors equal to 0.

    image8.png
  3. Solve for the values of x that satisfy both equations.

    image9.png

    So the solutions are all of the form x = 90° + 180°n or x = 60° + 180°n.

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