## 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.

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

Solution:

To solve for x, first isolate the cosine term.

If we restrict the domain of the cosine function to , we can use the arccos function to solve for x.

The cosine is negative in the second and third quadrant. The period of this function is . Divide the interval from 0 to into four equal intervals representing quadrants: The cosine is negative in the interval and the solution is The cosine is also negative in the interval and the solution is

Since the period is this means that the values will repeat every radians. Therefore, the exact solutions are and 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 13.9393 for x, then 13.9393 is a solution.

Left Side:

Right Side:

Since the left side equals the right side when you substitute 8.0518339for x, then 8.0518339 is a solution.

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 8.05183396 (one of the solutions) as well as 13.9393. Since the period of the function is , the graph crosses again at 8.05183396+21.991149 = 30.04298 and 13.9393 + 21.991149 = 35.930449, etc.

If you would like to go back to the previous section, click on Previous.

If you would like to test yourself by working some problems similar to this example, click on problem.

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

Author: Nancy Marcus