## 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 9.3b:

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

The cosine of 6x is positive in the first quadrant and in the fourth quadrant .

This means that there are two solutions in the first counterclockwise rotation from 0 to . One angle, terminates in the first quadrant and angle terminates in the fourth quadrant.

Since the period is this means that the values will repeat every radians. Therefore, the 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 0.10725 for x, then 0.10725 is a solution.

Left Side:

Right Side:

Since the left side equals the right side when you substitute 0.9399474for x, then 0.9399474 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 0.10725 (one of the solutions). Since the period of the function is , the graph crosses again at 0.10725+1.04719755=1.15444755 and again at , etc.

Note the graph also crosses at 0.9399474 (one of the solutions). Since the period of the function is , the graph crosses again at 0.9399474+1.04719755=1.987145 and again at , etc.

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