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SOLVING TRIGONOMETRIC EQUATIONS

Note:

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

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

**
Example 3: **

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

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

The cosine is positive in the first quadrant and the fourth quadrant. This
means that one solution is an angle that terminates in the first quadrant
and one solution is an angle that terminates in the fourth quadrant. The
second solution is

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

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 answer
*x*=0.541099525957

- Left Side:

- Right Side: 0

Since the left side equals the right side when you substitute
0.541099525957 for x, then
0.541099525957 is a solution.

Check answer .
*x*=5.74208578

- Left Side:

- Right Side: 0

Since the left side equals the right side when you substitute
5.74208578for x, then
5.74208578 is a solution.

**Graphical Check:**

Graph the equation

Note that the graph crosses the
x-axis many times indicating many solutions. Two of the x-intercepts are
located at
0.541099525957 and
5.74208578. This means that these are two
solutions. Notice also that there is an x-intercept at
6.8244284833 which
is equal to
There is also an x-intercept at
12.02527108 which is equal to

**
If you would like to work another example, click on Example.
**

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

IF you would like to go to the next section, click on Next.

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