SOLVING TRIGONOMETRIC EQUATIONS

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

Problem 9.9d:        Solve for x in the equation

where n is an integer.

The approximate values of these solutions are

where n is an integer.

Solution:

There are an infinite number of solutions to this problem.

Convert to , substitute in the original equation, and simplify.

To solve for x, we have to isolate x. How do we isolate the x? We could take the inverse (arcsine) of both sides. However, inverse functions can only be applied to one-to-one functions and the sine 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 sine function is one-to-one on the interval If we restrict the domain of the sine function to that interval , we can take the arcsine of both sides of the equation and isolate the x.

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=0.7929528

Left Side:

Right Side:

Since the left side of the original equation equals the right side of the original equation when you substitute 0.7929528 for x, then 0.7929528 is a solution.

Check solution x=6.53743

Left Side:

Right Side:

Since the left side of the original equation equals the right side of the original equation when you substitute 6.53743 for x, then 6.53743 is a solution.

We have just verified algebraically that the exact solutions are and and these solutions repeat every units. The approximate values of these solutions are and 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 0.79295278. Since the period is , you can verify that the graph also crosses the x-axis again at and at etc.

Verify the graph crosses the x-axis at 6.53743. Since the period is , you can verify that the graph also crosses the x-axis again at and at 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 In this case, although there are an infinite number of solutions, only the solution is located in the interval

If you would like to review the solution of another problem, click on solution.

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.

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