 SOLVING TRIGONOMETRIC EQUATIONS Note: If you would like a review of trigonometry, click on trigonometry.

Example 1:        Solve for x in the following equation. There are an infinite number of solutions to this problem.

First isolate the cosine term. To solve for x, we have to isolate x. How do we isolate the x? We could take the inverse (arccosine) of both sides. However, inverse functions can only be applied to one-to-one functions and the cosine function is not one-to-one.

Let's restrict the domain so the function is one-to-one on the restricted domain while preserving the original range. The cosine function is one-to-one on the interval If we restrict the domain of the cosine function to that interval , we can take the arccosine of both sides of each equation.  The angle x is the reference angle. We know that Therefore, if , then  The period of equals and the period of equals , this means other solutions exists every units. The exact solutions are where n is an integer.

The approximate values of these solutions are where n is an integer.

You can check each solution algebraically by substituting each solution in the original equation. If, after the substitution, the left side of the original equation equals the right side of the original equation, the solution is valid.

You can also check the solutions graphically by graphing the function formed by subtracting the right side of the original equation from the left side of the original equation. The solutions of the original equation are the x-intercepts of this graph.

Algebraic Check:

Check solution x=0.8569321

Left Side: Right Side: Since the left side of the original equation equals the right side of the original equation when you substitute 0.8569321 for x, then 0.8569321 is a solution.

Check solution x=-0.19026544

Left Side: Right Side: Since the left side of the original equation equals the right side of the original equation when you substitute -0.19026544 for x, then -0.19026544is a solution.

We have just verified algebraically that the exact solutions are and these solutions repeat every units. The approximate values of these solutions are and 0.8569321 and these solutions repeat every units.

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. Let's check a few of these x-intercepts against the solutions we derived.

Verify the graph crosses the x-axis at -0.19026544. Since the period is , you can verify that the graph also crosses the x-axis again at and at , 7.997049, 10.091444 etc.

Verify the graph crosses the x-axis at 0.8569321. Since the period is , you can verify that the graph also crosses the x-axis again at and at , 7.1401171 etc.

Note: If the problem were to find the solutions in the interval , then you choose those solutions from the set of infinite solutions that belong to the set  1.90412966, 2.9513271, 5.0457221, and 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.

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[Calculus] [Complex Variables] [Matrix Algebra] S.O.S MATHematics home page Author: Nancy Marcus