If you would like an in-depth review of logarithms, the rules of logarithms, logarithmic functions and logarithmic equations, click on logarithmic functions.

Solve for x in the following equation.

Problem 9.2c:

$8\tan \left( \displaystyle \frac{1}{5}x\right) -14=0$

        There are an infinite number of solutions: $x=5\tan
^{-1}\left( \displaystyle \frac{7}{4}\right) \pm n\left( 5\pi \right) $ are the exact solutions, and $x\approx 5.25825\pm n\left( 5\pi \right) $ are the approximate solutions.


To solve for x, first isolate the tangent term.

\begin{eqnarray*}&& \\
8\tan \left( \displaystyle \frac{1}{5}x\right) -14 &=&0 ...
...tyle \frac{14}{8}=\displaystyle \frac{7}{4} \\
&& \\
&& \\

If we restrict the domain of the cosine function to $-\displaystyle \frac{\pi }{2}<\displaystyle \frac{
1}{5}x<\displaystyle \frac{\...
...}\rightarrow -\displaystyle \frac{5}{2}\leq x\leq \displaystyle \frac{5\pi }{2}$, we can use the arctan function to solve for x.

\begin{eqnarray*}\tan \left( \displaystyle \frac{1}{5}x\right) &=&\displaystyle ...
...ght) &=&\tan
^{-1}\left( \displaystyle \frac{7}{4}\right) \\
\displaystyle \frac{1}{5}x &=&\tan ^{-1}\left( \displayst...
... \frac{7}{4}\right) \\
&& \\
x &\approx &5.25825 \\
&& \\

Since the period is $5\pi ,$ this means that the values will repeat every $
5\pi $ radians. Therefore, the solutions are $x=x=5\tan ^{-1}\left( \displaystyle \frac{7
}{4}\right) \pm \pm n\left( 5\pi \right) $ $\approx 5.25825\pm n\left(
15.707963\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=5.25825

Since the left side equals the right side when you substitute 5.25825 for x, then 5.25825 is a solution.

Check the answer $x=5.25825+5\pi =20.966214$

Since the left side equals the right side when you substitute 20.966214for x, then 20.966214 is a solution.

Graphical Check:

Graph the equation

$f(x)=8\tan \left( \displaystyle \frac{1}{5}x\right) -14.$

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

Note the graph crosses at 5.25825 (one of the solutions). Since the period of the function is $5\pi \approx 15.70796$, the graph crosses again at 5.25825+15.70796=2.966214 and again at $
5.25825+2\left( 15.70796\right) \approx 36.674178$, etc.

If you would like to review the solution to problem 9.2d, click on solution.

If you would like to go back to the previous section, click on previous

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 contents.

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