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

Solve for x in the following equation.

Example 2:        

$2sin\left( x\right) -\displaystyle \frac{3}{5}=0$

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

\begin{eqnarray*}&& \\
2sin\left( x\right) -\displaystyle \frac{3}{5}=0 && \\
... && \\
&& \\
sin\left( x\right) =\displaystyle \frac{3}{10} &&

\begin{eqnarray*}&& \\
If\ we\ restrict\ the\ domain\ of\ the\ function\ to\ \l...
&& \\
&& \\
x\approx 0.304692654015 && \\
&& \\
&& \\

The sine function is positive in the first and second quadrant. If the reference angle is $x\approx 0.304692654015$, the angle that terminates in the second quadrant is $\pi -0.304692654015=2.83689999. $

The period of sin $\left( x\right) $ function is $2\pi .$ This means that the values will repeat every $2\pi $ radians. Therefore, the solutions are $
x=0.304692654015\pm n\left( 2\pi \right) $ and $x=2.836899999\pm n\left(
2\pi \right) $ 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 . x=0.304692654015

Since the left side equals the right side when you substitute 0.304692654015 for x, then 0.304692654015 is a solution.

Check answer . x=2.83689999957

Since the left side equals the right side when you substitute 2.83689999957for x, then 2.83689999957 is a solution.

Graphical Check:

Graph the equation

$f(x)=2\sin (x)-\displaystyle \frac{3}{5}.$

Note that the graph crosses the x-axis many times indicating many solutions.

Note that it crosses at (one of the solutions). Since the period of the function is $2\pi \approx 6.28318530718$, the graph crosses again at 2.83689999957+6.28318530718=9.12 and again at $
2.83689999957+2\left( 6.28318530718\right) \approx 15.4$, etc. The graph also crosses at 0.304692654015 (another solution we found). Since the period is $2\pi \approx 6.28318530718$, it will cross again at $0.304692654015+2\pi =6.58788$ and at $0.304692654015+2\left( 2\pi
\right) \approx 12.871$, 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.

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