SOLVING TRIGONOMETRIC EQUATIONS

Note:

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

Solve for the real number x in the following equation.

Problem 9.3c:         bf

Answers:        There are an infinite number of solutions: are the exact solutions, and are the approximate solutions.

Solution:

To solve for x, first isolate the tangent term.

If we restrict the domain of the tangent function to , we can use the arctan function to solve for x.

Since the period is this means that the values will repeat every radians. Therefore, the exact solutions are and the approximate solutions are 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=6.1388619

Left Side:

Right Side:

Since the left side equals the right side when you substitute 6.1388619for x, then 6.1388619 is a solution.bf

Check the answer .

Left Side:

Right Side:

Since the left side equals the right side when you substitute 21.846825for x, then 21.846825 is a solution.bf

Graphical Check: Graph the equation (Formed by subtracting the right side of the original equation from the left side of the original equation.

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

Note the graph crosses at 6.1388619 (one of the solutions). Since the period of the function is , the graph crosses again at 6.1388619+15.707963=21.846825 and again at , 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 back to the equation table of contents, click on Contents

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

Author: Nancy Marcus

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