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 9.1c:         

$\displaystyle \frac{1}{2}\tan \left( x\right) =20$

Answer:         There are an infinite number of solutions: $x=\tan ^{-1}\left(
40\right) \pm \pi $ are the exact solutions, and $x\approx 1.54580\pm \pi $are the approximate solutions.

Solution:         To solve for x, first isolate the tangent term.

\begin{eqnarray*}&& \\
\displaystyle \frac{1}{2}\tan \left( x\right) &=&20 \\
&& \\
\tan \left( x\right) &=&40 \\
&& \\
&& \\

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

\begin{eqnarray*}\tan \left( x\right) &=&40 \\
&& \\
\tan ^{-1}\left( \tan \le...
...}\left( 40\right) \\
&& \\
x &\approx &1.5458015 \\
&& \\

The period of tangent function is $\pi .$ This means that the values will repeat every $\pi $ radians. Therefore, the solutions are $x=1.5458015\pm
n\pi $ 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=1.5458015

Since the left side equals the right side when you substitute 1.5458015for x, then 1.5458015 is a solution.

Graphical Check:

Graph the equation

$f(x)=\displaystyle \frac{1}{2}\tan \left( x\right) -20.$

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

Note the graph crosses at 1.5458015 (one of the solutions). Since the period of the function is $\pi \approx 3.141592$, the graph crosses again at 1.5458015+3.1415924.687394 and again at $
0.38439677+2\left( 3.141592\right) \approx 7.8289868$, etc.

If you would like to review the solution to problem 9.1d, click on solution.

If you would like to go back to the equation table of contents, click on contents.

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