## SOLVING TRIGONOMETRIC EQUATIONS

Note:

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

Solve for x in the following equation.

Example 3:

There are an infinite number of solutions to this problem. To solve for x, you must first isolate the cosine term.

If we restriction the domain of the cosine function to , we can use the inverse cosine function to solve for reference angle and then x.

The period of is and the period of is Divide the interval into four equal intervals:

We know that the cosine function is positive in the first and the fourth quadrant (intervals). Therefore two of the solutions are the angle that terminates in the first quadrant and the angle that terminates in the fourth quadrant. We have already solved for

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 0.16991845for x, then 0.16991845 is a solution.

• Left Side:

• Right Side:

Since the left side equals the right side when you substitute 1.400878 for x, then 1.400878 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 0.16991845. Since the period is , it crosses again at 0.16991845+1.570796=1.74071477 and at 0.16991845+2(1.570796)=3.31151045, etc.

The graph also crosses the x-axis at 1.400878. Since the period is , it crosses again at 1.400878+1.570796=2.971674and at 1.400878+2(1.570796)=4.5424707, 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