SOLVING TRIGONOMETRIC EQUATIONS


Note:

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


Solve for x in the following equation.


Example 4:

$5\tan \left( \displaystyle \frac{x}{3}\right) -10=3\tan \left( \displaystyle \frac{x }{3}\right) $



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

\begin{eqnarray*}&& \\ 5\tan \left( \displaystyle \frac{x}{3}\right) -10 &=&3\t... ...&10 \\ && \\ \tan \left( \displaystyle \frac{x}{3}\right) &=&5 \end{eqnarray*}


If we restriction the domain of the tangent function to $\left( -\displaystyle \frac{\pi }{2}<\displaystyle \frac{x}{3}<\displaystyle \... ... \left( -\displaystyle \frac{3\pi }{2} <x<\displaystyle \frac{3\pi }{2}\right) $, we can use the inverse tangent function to solve for reference angle $x^{\prime }$ and then x.

\begin{eqnarray*}&& \\ \tan \left( \displaystyle \frac{x}{3}\right) &=&5 \\ &&... ...e \frac{x}{3}\right) \right) &=&\tan ^{-1}\left( 5\right) \\ && \end{eqnarray*}
\begin{eqnarray*}&&\\ \displaystyle \frac{x}{3} &=&\tan ^{-1}\left( 5\right) \\... ...tan ^{-1}\left( 5\right) \\ && \\ x &\approx &4.1202023 \\ && \end{eqnarray*}
\begin{eqnarray*}\mbox{ Reference Angle } &:&x^{\prime }\approx 3\tan ^{-1}\left... ...nce Angle } &:&x^{\prime }\approx 4.1202023 \\ && \\ && \\ && \end{eqnarray*}


The solution is $x=3\tan ^{-1}\left( 5\right) .\bigskip\bigskip\bigskip $

The period of the tangent function is $\pi $ and the period of this function is $3\pi .$ This means that the values will repeat every $3\pi $ radians in both directions. Therefore, the exact solutions are $x=3\tan ^{-1}\left( 5\right) \pm n\left( 3\pi \right) $ n is an integer.


The approximate solutions are $x\approx 4.1202023\pm n\left( 3\pi \right) .$



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=4.1202023


Since the left side equals the right side when you substitute 4.1202023for x, then 4.1202023 is a solution.




Check answer $x=4.1202023+3\pi \approx 13.54498$


Since the left side equals the right side when you substitute 13.54498 for x, then 13.54498 is a solution.




Graphical Check:


Graph the equation $f(x)=5\tan \left( \displaystyle \frac{x}{3}\right) -10-3\tan \left( \displaystyle \frac{x}{3}\right) =2\tan \left( \displaystyle \frac{x}{3}\right) -10.$ Note that the graph crosses the x-axis many times indicating many solutions.


The graph crosses the x-axis at 4.1202023. Since the period is $3\pi \approx 9.424778$, it crosses again at 4.1202023+9.424778=13.54498 and at 4.1202023+2(9.424778)=22.969758, etc.



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


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