## SOLVING TRIGONOMETRIC EQUATIONS

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

Problem 9.5b:         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. To solve for x, set the equation equal to zero and factor.

then

How do we isolate the x in the equations? We could take the arccosine of both sides of the cosine equation and the arcsine of both sides of the sine equations. However, the sine and cosine functions are not one-to-one functions.

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 cosine equation. The graph of 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 sine equation.

Since we know that if then and

Now solve the equations First we have

then we have

We know that Therefore, if then Solve the equations

and

Since the period of both the cosine function and the sine function is , the exact solutions are

where n is an integer.

The approximate values of these solutions are

where n is an integer.

These solutions may or may not be the answers to the original problem. You much check them, either numerically or graphically, with the original equation.

Numerical Check:

Left Side:

Right Side:

Since the left side equals the right side when you substitute 1.5707963for x, then 1.5707963 is a solution.

Left Side:

Right Side:

Since the left side equals the right side when you substitute -1.5707963for x, then -1.5707963 is a solution.

Left Side:

Right Side:

Since the left side equals the right side when you substitute 0.4636476 for x, then 0.4636476 is a solution.

Left Side:

Right Side:

Since the left side equals the right side when you substitute -0.4636476 for x, then -0.4636476 is a solution.

Left Side:

Right Side:

Since the left side equals the right side when you substitute 2.677945 for x, then 2.677945 is a solution.

Left Side:

Right Side:

Since the left side equals the right side when you substitute 3.60524 for x, then 3.60524 is a solution.

Graphical Check:

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

Verify that it crosses at 1.5707963. Since the period is , it crosses again at 1.5707963+6.2831853=7.8539816 and at 1.5707963-6.2831853=-4.712389, etc.

Verify that it crosses at -1.5707963. Since the period is , it crosses again at -1.5707963+6.2831853=4.712389 and at -1.5707963-6.2831853=-7.85398166, etc.

Verify that it crosses at 0.4636476. Since the period is , it crosses again at 0.4636476+6.2831853=6.7468329 and at 0.4636476-6.2831853=-5.8195376, etc.

Verify that it crosses at -0.4636476. Since the period is , it crosses again at -0.4636476+6.2831853=5.8195377 and at -0.4636476-6.2831853=-6.7468329, etc.

Verify that it crosses at 2.677945. Since the period is , it crosses again at 2.677945+6.2831853=8.96113035 and at 2.677945-6.2831853=-3.60524, etc.

Verify that it crosses at 3.60524. Since the period is , it crosses again at 3.60524+6.2831853=9.88842557 and at 3.60524-6.2831853=-2.677945, etc.

Note: If the problem were to find the solutions in the interval , then you would choose those solutions from the set of infinite solutions that belong to the set : , and 5.8195377.

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

Author: Nancy Marcus