SOLVING TRIGONOMETRIC EQUATIONS

Note:

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

Solve for x in the following equation.

Example 4:

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

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

The solution is

The period of the tangent function is and the period of this function is This means that the values will repeat every radians in both directions. Therefore, the exact solutions are n is an integer.

The approximate solutions are

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 4.1202023for x, then 4.1202023 is a solution.

• Left Side:

• Right Side:

Since the left side equals the right side when you substitute 13.54498 for x, then 13.54498 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 4.1202023. Since the period is , it crosses again at 4.1202023+9.424778=13.54498 and at 4.1202023+2(9.424778)=22.969758, etc.

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