SOLVING TRIGONOMETRIC EQUATIONS


Note:

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



Solve for x in the following equation.


Example 4:         

$\displaystyle \frac{3}{4}\tan \left( x\right) -\displaystyle \frac{7}{10}=0$

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

Isolate the tangent term.


\begin{eqnarray*}&& \\
\displaystyle \frac{3}{4}\tan \left( x\right) -\displays...
...e \frac{28}{30}=\displaystyle \frac{14}{15} \\
&& \\
&& \\
&&
\end{eqnarray*}


If we restrict the domain of the tangent function to $\left( -\displaystyle \frac{\pi }{2}
,\displaystyle \frac{\pi }{2}\right) $, we can use the arctan function to solve for x.

\begin{eqnarray*}\tan \left( x\right) &=&\displaystyle \frac{14}{15} \\
&& \\
...
...ystyle \frac{14}{15}\right) \approx 0.750929062398 \\
&& \\
&&
\end{eqnarray*}


The period of tan $\left( x\right) $ function is $\pi .$ This means that the values will repeat every $\pi $ radians. Therefore, the exact solutions are $
x=\tan ^{-1}\left( \displaystyle \frac{14}{15}\right) \pm n\left( \pi \right) .$ The approximate solutions are $x\approx 0.750929062398\pm n\left( \pi \right)
. $



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=0.750929062398

Since the left side equals the right side when you substitute 0.750929062398 for x, then 0.750929062398 is a solution.



Check answer . $x=0.750929062398+\pi \approx 3.89252171599$

Since the left side equals the right side when you substitute 3.89252171599for x, then 3.89252171599 is a solution.

Graphical Check:

Graph the equation

$f(x)=\displaystyle \frac{3}{4}\tan \left( x\right) -\displaystyle \frac{7}{10}.$

Note that the graph crosses the x-axis many times indicating many solutions. One of the x-intercepts is located at 0.750929062398. This means that this is a solution. Notice that the distance between each x-intercepts is $\pi .$



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


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