SOLVING TRIGONOMETRIC EQUATIONS

Note:

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

Solve for x in the following equation.

Example 3:        

$7\cos \left( x\right) -6=0$

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

$$\begin{eqnarray*}&& \\ 7\cos \left( x\right) -6 &=&0 \\ && \\ 7\cos \left( x\... ...left( x\right) &=&\displaystyle \frac{6}{7} \\ && \\ && \\ && \end{eqnarray*}$$

If we restrict the domain of the cosine function to $\left[ 0,\pi \right]$, we can use the $\arccos$ function to solve for x.

$$\begin{eqnarray*}\cos ^{-1}\left( \cos \left( x\right) \right) &=&\cos ^{-1}\lef... ...laystyle \frac{6}{7}\right) \approx 0.541099525957 \\ && \\ && \end{eqnarray*}$$

The cosine is positive in the first quadrant and the fourth quadrant. This means that one solution is an angle that terminates in the first quadrant and one solution is an angle that terminates in the fourth quadrant. The second solution is

$$\begin{eqnarray*}x &=&2\pi -\cos ^{-1}\left( \displaystyle \frac{6}{7}\right) \approx 5.74208578 \\ && \\ && \\ && \end{eqnarray*}$$

The period of cos $\left( x\right)$ function is $2\pi .$ This means that the values will repeat every $2\pi$ radians. Therefore, the exact solutions are $x=\cos ^{-1}\left( \displaystyle \frac{6}{7}\right) \pm n\left( 2\pi \right)$ and $x=\left( 2\pi -\cos ^{-1}\left( \displaystyle \frac{6}{7}\right) \right) \pm n\left( 2\pi \right)$ where n is an integer. The approximate solutions are $x\approx 0.541099525957\pm n\left( 2\pi \right) \$and $x\approx 5.74208578\pm n\left( 2\pi \right)$

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=0.541099525957

• Left Side: $\qquad 7\cos \left( x\right) -6=7\cos \left( 0.541099525957\right) -6\approx 0$

• Right Side:        0

Since the left side equals the right side when you substitute 0.541099525957 for x, then 0.541099525957 is a solution.

Check answer . x=5.74208578

• Left Side: $\qquad 7\cos \left( x\right) -6\approx 7\cos \left( 5.74208578\right) -6\approx 0$

• Right Side:        0

Since the left side equals the right side when you substitute 5.74208578for x, then 5.74208578 is a solution.

Graphical Check:

Graph the equation

$f(x)=7\cos (x)-6.$

Note that the graph crosses the x-axis many times indicating many solutions. Two of the x-intercepts are located at 0.541099525957 and 5.74208578. This means that these are two solutions. Notice also that there is an x-intercept at 6.8244284833 which is equal to $0.541099525957+2\pi .$ There is also an x-intercept at 12.02527108 which is equal to $5.74208578+2\pi .$

If you would like to work another example, click on Example.

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

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