SOLVING TRIGONOMETRIC EQUATIONS


Note:

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


Solve for x in the following equation.


Example 2:

$11\sin \left( x\right) +7=3\sin x$



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

\begin{eqnarray*}&& \\ 11\sin \left( x\right) +7 &=&3\sin x \\ && \\ 8\sin \l... ...eft( x\right) &=&-\displaystyle \frac{7}{8} \\ && \\ && \\ && \end{eqnarray*}


If we restrict the domain of the sine function to $\left[ -\displaystyle \frac{\pi }{2} \leq x\leq ,\displaystyle \frac{\pi }{2}\right] $, we can use the inverse sine function to solve for reference angle x, and then x.

\begin{eqnarray*}&& \\ \sin \left( x\right) &=&-\displaystyle \frac{7}{8} \\ &... ... x &=&\sin ^{-1}\left( -\displaystyle \frac{7}{8}\right) \\ && \end{eqnarray*}
\begin{eqnarray*}&&\\ x &\approx &-1.06543581651 \\ && \\ \mbox{ Reference An... ...Angle } &:&x^{\prime }\approx 1.06543581651 \\ && \\ && \\ && \end{eqnarray*}


We know that the $\sin $e function is negative in the third and the fourth quadrant. Therefore two of the solutions are the angle $x_{1}=\pi +x^{\prime }$ that terminates in the third quadrant and the angle $x_{2}=2\pi -x^{\prime }$ that terminates in the fourth quadrant. We have already solved for $x^{\prime }.$

\begin{eqnarray*}&& \\ x_{1} &=&\pi +x^{\prime }=\pi +\sin ^{-1}\left( \display... ...\right) \\ && \\ x_{2} &\approx &5.217749 \\ && \\ && \\ && \end{eqnarray*}

The solutions are $x=\pi +\sin ^{-1}\left( \displaystyle \frac{7}{8}\right) $ and $x=2\pi -\sin ^{-1}\left( \displaystyle \frac{7}{8}\right) .\bigskip\bigskip\bigskip $

The period of the sin $\left( x\right) $ function is $2\pi .$ This means that the values will repeat every $2\pi $ radians in both directions. Therefore, the exact solutions are $x=\pi +\sin ^{-1}\left( \displaystyle \frac{7}{8}\right) \pm n\left( 2\pi \right) $ and $x=2\pi -\sin ^{-1}\left( \displaystyle \frac{7}{8}\right) \pm n\left( 2\pi \right) $ where n is an integer.


The approximate solutions are $x\approx 4.207028\pm n\left( 2\pi \right) $and $x\approx 5.217749\pm \pm n\left( 2\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 answer x=4.207028


Since the left side equals the right side when you substitute 4.207028 for x, then 4.207028 is a solution.




Check answer x=5.217749


Since the left side equals the right side when you substitute 5.217749 for x, then 5.217749 is a solution.



Graphical Check:


Graph the equation

$f(x)=11\sin \left( x\right) +7-3\sin x=8\sin \left( x\right) +7.$

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


The graph crosses the x-axis at 4.207028. Since the period is $2\pi \approx 6.2831853$, it crosses again at 4.207028+6.2831853=10.4902133 and at 4.207028+2(6.2831853)=16.7733986, etc.


The graph also crosses the x-axis at 5.217749. Since the period is $2\pi \approx 6.2831853$, it crosses again at 5.217749+6.2831853=11.50093 and at 5.217749+2(6.2831853)=17.78411, etc.


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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