SOLVING TRIGONOMETRIC EQUATIONS

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

Problem 9.7b:        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.

In the equation there are two different trigonometric terms. The problem will be less difficult to solve if you convert the equation to an equivalent equation with just sine terms or just cosine terms.

Which one shall it be? The easiest way is to convert the trigonometric terms to sine terms because from the identity , you can solve for . Substitute for in the original equation.

Rewrite the equation in factored form and solve.

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

We have transformed a difficult problem into two less difficult problems. To find the solutions to the original equation, , we first solve for in the in the equations and

and

To solve for x, we must first isolate the x, and just how do we isolate the x? We could take the arcsine of both sides. However, the sine function is not a one-to-one function.

We could restrict the domain of the function so that 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 , 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.

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 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:

Since the left side of the original equation equals the right side of the original equation when you substitute 0.52359878 for x, then 0.52359878is 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 2.6179939 for x, then 2.6179939 is a solution.

The exact solutions are and and these solutions repeat every units. The approximate values of these solutions are and 2.6179939 and these solutions repeat every units.

Graphical Check:

Graph the equation Note that the graph crosses the x-axis many times indicating many solutions.

Verify the graph crosses the x-axis at 0.52359878. Since the period is , the graph also crosses the x-axis again at 0.52359878+6.2831853=6.806784 and at , etc.

Verify the graph crosses the x-axis at . Since the period is , the graph also crosses the x-axis again at 2.6179939+6.2831853=8.901179 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

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Author: Nancy Marcus