## SOLVING TRIGONOMETRIC EQUATIONS

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

Problem 9.12a:        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. Let's simplify the problem by rewriting it in an equivalent factored form.

The only way the product equals zero is if at least one of the factors equals zero. Therefore, if , or

How do we isolate the x in each of these equations? We could take the inverse (arcsine) of both sides of the equation. However, the sine function is not a one-to-one function.

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 each equation.

We know that Therefore, if .

Since the period of equals , these solutions will repeat 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

Left Side:

Right Side:

Since the left side of the original equation equals the right side of the original equation when you substitute 0.72972766 for x, then 0.72972766is a solution.

Check solution

Left Side:

Right Side:

Since the left side of the original equation equals the right side of the original equation when you substitute 2.411865 for x, then 2.411865 is a solution.

Check solution

Left Side:

Right Side:

Since the left side of the original equation equals the right side of the original equation when you substitute 0.442911 for x, then 0.442911 is a solution.

Check solution

Left Side:

Right Side:

Since the left side of the original equation equals the right side of the original equation when you substitute 2.411865 for x, then 2.411865 is a solution.

Check solution

Left Side:

Right Side:

Since the left side of the original equation equals the right side of the original equation when you substitute 3.394273 for x, then 0.72972766 is a solution.

Check solution

Left Side:

Right Side:

Since the left side of the original equation equals the right side of the original equation when you substitute 2.69868 for x, then 2.69868 is 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 <tex2htmlcommentmark> 2.69868 and these solutions repeat every units.

Graphical Check:

Graph the function , formed by subtracting the right side of the original equation from the left side of the original equation. The x-intercepts are the real solutions.

. 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 -0.252680. Since the period is , you can verify that the graph also crosses the x-axis again at -0.252680+6.2831853=6.0305053 and at , etc.

Verify the graph crosses the x-axis at 0.442911. Since the period is , you can verify that the graph also crosses the x-axis again at 0.442911+6.2831853=6.7260963 and at 13.009282, etc.

Verify the graph crosses the x-axis at 0.729728. Since the period is , you can verify that the graph also crosses the x-axis again at 0.729728+6.2831853=7.0129133 and at etc.

Verify the graph crosses the x-axis at 2.411865. Since the period is , you can verify that the graph also crosses the x-axis again at 2.411865+6.2831853=8.6950503 and at , etc.

Verify the graph crosses the x-axis at 2.69868. Since the period is , you can verify that the graph also crosses the x-axis again at 2.69868+6.2831853=8.9818653 and at etc.

Verify the graph crosses the x-axis at 3.394273. Since the period is , you can verify that the graph also crosses the x-axis again at 3.394273+6.2831853=9.6774583 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 and

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

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

Author: Nancy Marcus