##
SOLVING TRIGONOMETRIC EQUATIONS

Note:

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

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

**
Example 1:**

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

We know that the
therefore
The sine function is positive in quadrants I
and II. The
is also equal to
Therefore, two of the solutions to the
problem 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 answer .

- Left Side:

- Right Side: 0

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

Check answer .

- Left Side:

- Right Side: 0

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

**Graphical Check:**

Graph the equation

*f* (*x*) = 2 sin(*x*) - 1

Note that the graph crosses the
x-axis many times indicating many solutions. Note that it crosses at
.
Since the period is
,
it
crosses again at
0.5236+6.283=6.81 and at
0.5236+2(6.283)=13.09, etc.
The graph crosses at
.
Since the period is
,
it will cross again at
2.618+6.283=8.9011 and at
2.618+2(6.283)=15.18, etc.

**
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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