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SOLVING TRIGONOMETRIC EQUATIONS

Note:

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

**Solve for the real number x in the following equation.**

Problem 9.3b:

Answers: There are an infinite number of solutions:
and
are the exact solutions, and
and
are the approximate solutions.

Solution:

To solve for x, first isolate the cosine term.

If we restrict the domain of the cosine function to
,
we can use the arccos function to
solve for x.

The period of
is
The period of
is
Divide the interval
into four equal intervals:
and

The cosine of 6x is positive in the first quadrant
and in the fourth quadrant
.

This means that there are two solutions in the first counterclockwise
rotation from 0 to
.
One angle,
terminates in the first
quadrant and angle
terminates in the fourth quadrant.

Since the period is
this means that the values will
repeat every
radians. Therefore, the solutions are
and
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:

Check the answer .*x*=0.10725

Left Side:

Right Side:

Since the left side equals the right side when you substitute 0.10725 for
x, then 0.10725 is a solution.

Check the answer .
*x*=0.9399474

Left Side:

Right Side:

Since the left side equals the right side when you substitute 0.9399474for x, then 0.9399474 is a solution.

**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.

Note the graph crosses at 0.10725 (one of the solutions). Since the period of the function is
,
the graph crosses again at
0.10725+1.04719755=1.15444755 and
again at
,
etc.

Note the graph also crosses at 0.9399474 (one of the solutions). Since the period of the function is
,
the graph crosses again at
0.9399474+1.04719755=1.987145 and
again at
,
etc.

**If you would like to test yourself by working some problems similar to this
example, click on problem.**

**
**

**
If you would like to go back to the equation table of contents, click on
Contents**

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[Algebra]
[Trigonometry]
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[Geometry]
[Differential Equations]
[Calculus]
[Complex Variables]
[Matrix Algebra]
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Author:
Nancy Marcus

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