##
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.**

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 cosine is negative in the second and third quadrant. The period of this
function is .
Divide the interval from 0 to
into four equal
intervals representing quadrants:
The cosine is
negative in the interval
and the solution is
The cosine is also negative in
the interval
and the
solution is

Since the period is
this means that the values will repeat every
radians. Therefore, the exact solutions are
and
The approximate 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*=13.9393

Left Side:

Right Side:

Since the left side equals the right side when you substitute 13.9393 for
x, then 13.9393 is a solution.

Check the answer .
*x*=8.0518339

Left Side:

Right Side:

Since the left side equals the right side when you substitute 8.0518339for x, then 8.0518339 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
8.05183396 (one of the solutions) as well as 13.9393. Since the period of the function is
,
the
graph crosses again at
8.05183396+21.991149 = 30.04298 and 13.9393 + 21.991149 = 35.930449, etc.

If you would like to go back to the previous section, click on Previous.

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

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