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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 9.3c:
bf

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

Solution:

To solve for x, first isolate the tangent term.

If we restrict the domain of the tangent function to
,
we can use the arctan function to solve for x.

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

Check the answer .
*x*=6.1388619

Left Side:

Right Side:

Since the left side equals the right side when you substitute 6.1388619for x, then 6.1388619 is a solution.bf

Check the answer .

Left Side:

Right Side:

Since the left side equals the right side when you substitute 21.846825for x, then 21.846825 is a solution.bf

**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 6.1388619 (one of the solutions). Since the period of the function is
,
the
graph crosses again at
6.1388619+15.707963=21.846825 and again at
,
etc.

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

**
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Author:
Nancy Marcus

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