SOLVING TRIGONOMETRIC EQUATIONS

Note:

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

Solve for x in the following equation.

Example 4:         

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

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

• Left Side: $\qquad \displaystyle \frac{3}{4}\tan \left( x\right) -\displaystyle \frac{7}{10... ...c{3}{ 4}\tan \left( 0.750929062398\right) -\displaystyle \frac{7}{10}\approx 0$

• Right Side:        0

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

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

• Left Side: $\qquad \displaystyle \frac{3}{4}\tan \left( x\right) -\displaystyle \frac{7}{10... ...ac{3}{ 4}\tan \left( 3.89252171599\right) -\displaystyle \frac{7}{10}\approx 0$

• Right Side:        0

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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