## SOLVING TRIGONOMETRIC EQUATIONS Note:

If you would like an review of trigonometry, click on trigonometry.

Solve for x in the following equation.

Example 2: There are an infinite number of solutions to this problem. To solve for x, you must first isolate the sine term. If we restriction the domain of the sine function to , we can use the inverse sine function to solve for reference angle 5x and then x.  We know that the e function is positive in the first and the second quadrant. Therefore two of the solutions are the angle 5x that terminates in the first quadrant and the angle that terminates in the second quadrant. We have already solved for 5x.  The exact solutions are and The period of the sin function is This means that the values will repeat every radians in both directions. Therefore, the exact solutions are and where n is an integer.

The approximate solutions are and where n is an integer.

These solutions may or may not be the answers to the original problem. You much check them, either numerically or graphically, with the original equation.

Numerical Check:

Check the answer x=0.028669513781

• Left Side: • Right Side: Since the left side equals the right side when you substitute 0.028669513781 for x, then 0.028669513781 is a solution.

Check the answer x=0.599649016937

• Left Side: • Right Side: Since the left side equals the right side when you substitute for x, then is a solution.

Graphical Check:

Graph the equation Note that the graph crosses the x-axis many times indicating many solutions.

Note that it crosses at 0.028669513781. Since the period is , it crosses again at 0.028669513781+1.256637=1.285306and at 0.028669513781+2(1.256637)=2.54194, etc.

The graph crosses at 0.599649016937. Since the period is , it will cross again at and at 0.599649016937+2(1.256637)=3.112923, etc If you would like to work another example, click on Example.

If you would like to test yourself by working some problems similar to this example, click on Problem.

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