##
SOLVING TRIGONOMETRIC EQUATIONS

Note:

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

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

**
Example 4:**

There are an infinite number of solutions to this problem. To solve for x,
you must first isolate the sine term.

If we restriction the domain of the sine function to
,
we can use the inverse
sine function to solve for reference angle
and then *x*.
We know that the e function is positive in the first and the second
quadrant. Therefore two of the solutions are the angle
that
terminates in the first quadrant and the angle
that
terminates in the second quadrant. We have already solved for

The exact solutions are
and

The period of the sin
function is
This means that the values will repeat every
radians in both directions. Therefore, the exact solutions are
and
where n is an integer.

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*=0.131705318835

- Left Side:

- Right Side:

Since the left side equals the right side when you substitute
0.131705318835 for x, then
0.131705318835 is a solution.

Check the answer
*x*=1.29629134189

- Left Side:

- Right Side:

Since the left side equals the right side when you substitute
1.29629134189for x, then
1.29629134189 is a solution.

**Graphical Check:**

Graph the equation

Note that the graph crosses the x-axis many times indicating many solutions.
Note that it crosses at
0.131705318835. Since the period is
,
it crosses again at
0.131705318835 + 2.85599 = 2.98769864028 and at
0.131705318835 + 2( 2.85599 ) = 5.843685, etc.

The graph crosses at
1.29629134189. Since the period is
,
it will cross again at
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
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