SOLVING TRIGONOMETRIC EQUATIONS

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Solve for the real number x in the following equation.

$5\cos \left( \displaystyle \frac{2}{7}x\right) +8=-\cos \left( \displaystyle \frac{2}{7}x\right) +4$

Answers:        There are an infinite number of solutions: $x=\displaystyle \frac{7\pi }{2} -\displaystyle \frac{7}{2}\cos ^{-1}\left( \displaystyle \frac{2}{3}\right) \pm n\left( 7\pi \right)$and $x=\displaystyle \frac{7\pi }{2}+\displaystyle \frac{7}{2}\cos ^{-1}\left( \displaystyle \frac{2}{3}\right) \pm n\left( 7\pi \right)$ are the exact solutions, and $x\approx 8.0518339\pm n\left( 7\pi \right)$ and $13.9393\pm n\left( 7\pi \right)$ are the approximate solutions.

Solution:

To solve for x, first isolate the cosine term.

$$\begin{eqnarray*}&& \\ 5\cos \left( \displaystyle \frac{2}{7}x\right) +8 &=&-\c... ... \frac{2}{7}x\right) &=&-\displaystyle \frac{2}{3} \\ && \\ && \end{eqnarray*}$$

If we restrict the domain of the cosine function to $0\leq \displaystyle \frac{2x}{7}\leq \pi \rightarrow 0\leq x<\displaystyle \frac{7\pi }{2}$, we can use the arccos function to solve for x.

$$\begin{eqnarray*}\cos \left( \displaystyle \frac{2}{7}x\right) &=&-\displaystyle... ...c{7}{2}\cos ^{-1}\left( -\displaystyle \frac{2}{3}\right) \\ && \end{eqnarray*}$$

$$\begin{eqnarray*}&&\\ \mbox{ Reference Angle } &:&x^{\prime }=\displaystyle \fr... ...{ Reference Angle } &:&x^{\prime }\approx 2.943736 \\ && \\ && \end{eqnarray*}$$

The cosine is negative in the second and third quadrant. The period of this function is $7\pi$. Divide the interval from 0 to $7\pi$ into four equal intervals representing quadrants: $\left[ 0,\displaystyle \frac{7\pi }{4}\right] ,\ \left[ \displaystyle \frac{7\... ...frac{21\pi }{4}\right] ,\ \left[ \displaystyle \frac{21\pi }{4},7\pi \right] .$ The cosine is negative in the interval $\ \left[ \displaystyle \frac{7\pi }{4},\displaystyle \frac{7\pi }{2}\right] ,$and the solution is $x_{1}=\displaystyle \frac{7\pi }{2}-x^{\prime }=\displaystyle \frac{7\pi }{2}-\displaystyle \frac{ 7}{2}\cos ^{-1}\left( \displaystyle \frac{2}{3}\right) .$ The cosine is also negative in the interval $\left[ \displaystyle \frac{7\pi }{2},\displaystyle \frac{21\pi }{4}\right] ,$ and the solution is $x_{2}=\displaystyle \frac{7\pi }{2}+x^{\prime }=\displaystyle \frac{7\pi }{2}+\displaystyle \frac{7}{2} \cos ^{-1}\left( \displaystyle \frac{2}{3}\right)$

Since the period is $7\pi ,$ this means that the values will repeat every $7\pi$ radians. Therefore, the exact solutions are $x=\displaystyle \frac{7\pi }{2} -\displaystyle \frac{7}{2}\cos ^{-1}\left( \displaystyle \frac{2}{3}\right) \pm n\left( 7\pi \right)$ and $x= \displaystyle \frac{7\pi }{2}+\displaystyle \frac{7}{2}\cos ^{-1}\left( \displaystyle \frac{2}{3}\right) \pm n\left( 7\pi \right) .$ The approximate solutions are $x\approx 8.0518339\pm n\left( 7\pi \right)$and $13.9393\pm n\left( 7\pi \right)$ 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=13.9393

Left Side: $\qquad 5\cos \left( \displaystyle \frac{2}{7}x\right) +8\approx 5\cos \left( \displaystyle \frac{2}{7}\left( 13.9393\right) \right) +8\approx 4.666666$

Right Side:         $-\cos \left( \displaystyle \frac{2}{7}x\right) +4\approx -\cos \left( \displaystyle \frac{2}{7}\left( 13.9393\right) \right) +4\approx 4.666666$

Since the left side equals the right side when you substitute 13.9393 for x, then 13.9393 is a solution.

Check the answer . x=8.0518339

Left Side: $\qquad 5\cos \left( \displaystyle \frac{2}{7}x\right) +8\approx 5\cos \left( \displaystyle \frac{2}{7}\left( 8.0518339\right) \right) +8\approx 4.666666$

Right Side:         $-\cos \left( \displaystyle \frac{2}{7}x\right) +4\approx -\cos \left( \displaystyle \frac{2}{7}\left( 8.0518339\right) \right) +4\approx 4.666666$

Since the left side equals the right side when you substitute 8.0518339for x, then 8.0518339 is a solution.

Graphical Check: Graph the equation $f(x)=6\cos \left( \displaystyle \frac{2}{7}x\right) +4.$ (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 8.05183396 (one of the solutions) as well as 13.9393. Since the period of the function is $7\pi \approx 21.991149$, the graph crosses again at 8.05183396+21.991149 = 30.04298 and 13.9393 + 21.991149 = 35.930449, etc.

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