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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 9.3a:
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**
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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 sine term.

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

The sine of x is negative in the third quadrant and the fourth quadrant.
This means that there are two solutions in the first counterclockwise
rotation from 0 to .

One angle,
terminates in the third 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*=3.58450369766

Left Side:

Right Side:

Since the left side equals the right side when you substitute
3.58450369766for x, then
3.58450369766 is a solution.**
**

**
**

**
**

**
**

Check the answer .
*x*=5.840274

Left Side:

Right Side:

Since the left side equals the right side when you substitute 5.840274 for
x, then 5.840274 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 3.584503697667 ( one of the solutions ). Since the period of the function is
,
the
graph crosses again at
3.584503697667+6.2831853=9.867689 and again at
,
etc.

The graph also crosses at 5.840274 ( another solution we found ). Since the period is
,
it will cross
again at
5.840274+6.2831853=12.123459 and at
,
etc

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

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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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