## 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 restrict the domain of the sine function to , we can use the inverse sine function to solve for reference angle x, and then x.

We know that the e function is negative in the third and the fourth quadrant. Therefore two of the solutions are the angle that terminates in the third quadrant and the angle that terminates in the fourth quadrant. We have already solved for

The 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:

• Left Side:

• Right Side:

Since the left side equals the right side when you substitute 4.207028 for x, then 4.207028 is a solution.

• Left Side:

• Right Side:

Since the left side equals the right side when you substitute 5.217749 for x, then 5.217749 is a solution.

Graphical Check:

Graph the equation

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

The graph crosses the x-axis at 4.207028. Since the period is , it crosses again at 4.207028+6.2831853=10.4902133 and at 4.207028+2(6.2831853)=16.7733986, etc.

The graph also crosses the x-axis at 5.217749. Since the period is , it crosses again at 5.217749+6.2831853=11.50093 and at 5.217749+2(6.2831853)=17.78411, 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.

IF you would like to go to the next section, click on Next.

This site was built to accommodate the needs of students. The topics and problems are what students ask for. We ask students to help in the editing so that future viewers will access a cleaner site. If you feel that some of the material in this section is ambiguous or needs more clarification, please let us know by e-mail.

[Algebra] [Trigonometry]
[Geometry] [Differential Equations]
[Calculus] [Complex Variables] [Matrix Algebra]

Author: Nancy Marcus