## SOLVING TRIGONOMETRIC EQUATIONS

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

Example 2:        Solve for x in the following equation.

There are an infinite number of solutions to this problem.

Isolate the cosine term. To do this rewrite the left side of the equation in an equivalent factored form.

The product of two factors equals zero if at least one of the factors equals zeros. This means that if or

We just transformed a difficult problem into two easier problems. To find the solutions to the original equation, , we find the solutions to the equations and

and

How do we isolate the x? We could take the arccosine of both sides. However, the cosine 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 graph of 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.

We know that cos Therefore, if , then

We complete the problem by solving for the second factor.

We know that cos Therefore, if , then

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.

One 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.

One can also check the solutions graphically by graphing the function formed by the left side of the original equation and graphing the function formed by the right side of the original equation. The x-coordinates of the points of intersection are the solutions. The right side of the equation is 0 and f(x)=0 is the x-axis. So really what you are looking for are the x-intercepts to the function formed by the left side of the equation.

Algebraic Check:

Check solution

Left Side:

Right Side:        0

Since the left side of the original equation equals the right side of the original equation when you substitute 0.7227342478 for x, then <tex2htmlcommentmark> 0.7227342478 is a solution.

Check solution

Left Side:

Right Side:        0

Since the left side of the original equation equals the right side of the original equation when you substitute -0.7227342478 for x, then <tex2htmlcommentmark> -0.7227342478 is a solution.

Check solution

Left Side:

Right Side:        0

Since the left side of the original equation equals the right side of the original equation when you substitute 1.04719755 for x, then 1.04719755 is a solution.

Check solution

Left Side:

Right Side:        0

Since the left side of the original equation equals the right side of the original equation when you substitute -1.04719755 for x, then -1.04719755 is a solution.

We have just verified that and are the exact solutions 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 Note that the graph crosses the x-axis many times indicating many solutions.

The graph crosses the x-axis at 0.7227342478. Since the period is , you can verify that the graph also crosses the x-axis again at 0.7227342478+6.2831853=7.005919555 and at , etc.

The graph crosses the x-axis at -0.7227342478. Since the period is , you can verify that the graph also crosses the x-axis again at -0.7227342478+6.2831853=5.560451 and at , etc.

The graph crosses the x-axis at 1.04719755. Since the period is , you can verify that the graph also crosses the x-axis again at 1.04719755+6.2831853=7.7220282 and at , etc.

The graph crosses the x-axis at -1.04719755. Since the period is , you can verify that the graph also crosses the x-axis again at -1.04719755+6.2831853=5.2359878 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

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

Author: Nancy Marcus