SOLVING TRIGONOMETRIC EQUATIONS

Note:

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

Solve for x in the following equation.

Example 1:

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 3x 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 3x that terminates in the first quadrant and the angle that terminates in the second quadrant. We have already solved for 3x.

The solutions are and

The period of the 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 0.174532925for x, then 0.174532925 is a solution.

• 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. You can see that the graph crosses at 0.174532925. Since the period is , it crosses again at 0.174532925+2.094395=2.2689 and at 0.174532925+2(2.094395)=4.3633, etc. The graph crosses at 0.872665.

Since the period is , it will cross again at and at 0.872665+2(2.094395)=5.061455, 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.

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Author: Nancy Marcus