SOLVING TRIGONOMETRIC EQUATIONS

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Solve for the real number x in the following equation.

Problem 9.3c:         $8\tan \left( \displaystyle \frac{1}{5}x\right) -14=3\tan \left( \displaystyle \frac{1}{5}x\right)$bf

Answers:        There are an infinite number of solutions: $x=5\tan ^{-1}\left( \displaystyle \frac{14}{5}\right) \pm n\left( 5\pi \right)$ are the exact solutions, and $x\approx 6.1388619\pm n\left( 5\pi \right)$ are the approximate solutions.

Solution:

To solve for x, first isolate the tangent term.

$$\begin{eqnarray*}&& \\ 8\tan \left( \displaystyle \frac{1}{5}x\right) -14 &=&3\... ... \frac{1}{5}x\right) &=&\displaystyle \frac{14}{5} \\ && \\ && \end{eqnarray*}$$

If we restrict the domain of the tangent function to $-\displaystyle \frac{\pi }{2}< \displaystyle \frac{x}{5}<\displaystyle \frac{\pi }{2}\rightarrow -\displaystyle \frac{5\pi }{2}<x<\displaystyle \frac{5\pi }{2}$, we can use the arctan function to solve for x.

$$\begin{eqnarray*}\tan \left( \displaystyle \frac{1}{5}x\right) &=&\displaystyle ... ... x &=&5\tan ^{-1}\left( \displaystyle \frac{14}{5}\right) \\ && \end{eqnarray*}$$

$$\begin{eqnarray*}&&\\ \mbox{ Reference Angle } &:&x^{\prime }=5\tan ^{-1}\left(... ... Reference Angle } &:&x^{\prime }\approx 6.1388619 \\ && \\ && \end{eqnarray*}$$

Since the period is $5\pi ,$ this means that the values will repeat every $5\pi$ radians. Therefore, the exact solutions are $x\approx 5\tan ^{-1}\left( \displaystyle \frac{14}{5}\right) \pm n\left( 5\pi \right)$ and the approximate solutions are $x\approx 6.1388619\pm n\left( 5\pi \right)$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 the answer . x=6.1388619

Left Side: $\qquad 8\tan \left( \displaystyle \frac{1}{5}x\right) -14=8\tan \left( \displaystyle \frac{ 1}{5}\left( 6.1388619\right) \right) -14\approx 8.399999$

Right Side:         $3\tan \left( \displaystyle \frac{1}{5}x\right) \approx 3\tan \left( \displaystyle \frac{1}{5}\left( 6.1388619\right) \right) \approx 8.399999\bigskip$

Since the left side equals the right side when you substitute 6.1388619for x, then 6.1388619 is a solution.bf

Check the answer . $x=6.1388619+5\pi =21.846825$

Left Side: $\qquad 8\tan \left( \displaystyle \frac{1}{5}x\right) -14=8\tan \left( \displaystyle \frac{ 1}{5}\left( 21.846825\right) \right) -14\approx 8.399999$

Right Side:         $3\tan \left( \displaystyle \frac{1}{5}x\right) \approx 3\tan \left( \displaystyle \frac{1}{5}\left( 6.1388619\right) \right) \approx 8.399999\bigskip$

Since the left side equals the right side when you substitute 21.846825for x, then 21.846825 is a solution.bf

Graphical Check: Graph the equation $f(x)=5\tan \left( \displaystyle \frac{1}{5}x\right) -14.$ (Formed by subtracting the right side of the original equation from the left side of the original equation.

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

Note the graph crosses at 6.1388619 (one of the solutions). Since the period of the function is $5\pi \approx 15.707963$, the graph crosses again at 6.1388619+15.707963=21.846825 and again at $6.1388619+2\left( 15.707963\right) \approx 37.554788$, etc.

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

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