SOLVING TRIGONOMETRIC EQUATIONS


Note: If you would like a review of trigonometry, click on trigonometry.


Example 3:        Solve for x in the following equation.



\begin{displaymath}11\sin \left( \displaystyle \displaystyle \frac{3x}{10}\right) +4=1\end{displaymath}


There are an infinite number of solutions to this problem.



First isolate the sine term.

\begin{displaymath}\begin{array}{rclll}
11\sin \left( \displaystyle \displaystyl...
...x}{10}\right) &=&-\displaystyle \frac{3}{11} \\
&&
\end{array}\end{displaymath}

To solve for x, we have to isolate x. How do we isolate the x? We could take the inverse (arcsine) of both sides. However, inverse functions can only be applied to one-to-one functions and the sine function is not one-to-one.


Let's restrict the domain so the function is one-to-one on the restricted domain while preserving the original range. The sine function is one-to-one on the interval $\left[ -\displaystyle \displaystyle \frac{\pi }{2},\displaystyle \displaystyle \frac{\pi }{2}\right] .$ If we restrict the domain of the sine function to that interval , we can take the arcsine of both sides of the equation and isolate the x.


\begin{displaymath}\begin{array}{rclll}
&& \\
-\displaystyle \displaystyle \fra...
...rac{3}{11}\right) \approx -1.01564218 \\
&& \\
&&
\end{array}\end{displaymath}

The angle x is the reference angle. We know that

\begin{displaymath}\begin{array}{rclll}
&& \\
\sin \left( \displaystyle \frac{3...
...t( \pi -\displaystyle \frac{3x}{10}\right) .
\\
&&
\end{array}\end{displaymath}

Therefore, if $\sin \left( \displaystyle \displaystyle \frac{3x}{10}\right) =-\displaystyle \displaystyle \frac{3}{11}$, then $%
\sin \left( \pi -\displaystyle \displaystyle \frac{3x}{10}\right) =-\displaystyle \displaystyle \frac{3}{11}.$


\begin{displaymath}\begin{array}{rclll}
&& \\
\sin \left( \pi -\displaystyle \d...
...ac{3}{11}\right) \right)
\approx 11.39273095 \\
&&
\end{array}\end{displaymath}



The period of $\sin (x)$ equals $2\pi $ and the period of $\sin \left(
\displaystyle \displaystyle \frac{3x}{10}\right) $ equals $\displaystyle \displaystyle \frac{20\pi }{3}$, this means other solutions exists every $\pm \displaystyle \displaystyle \frac{20\pi }{3}$ units. The exact solutions are

\begin{displaymath}\begin{array}{rclll}
&& \\
x_{1} &=&\displaystyle \frac{10}{...
...displaystyle \displaystyle \frac{20n\pi }{3} \\
&&
\end{array}\end{displaymath}

where n is an integer.




The approximate values of these solutions are

\begin{displaymath}\begin{array}{rclll}
&& \\
x_{1} &\approx &-0.9207554\pm 20....
...
x_{2} &\approx &11.39273095\pm 20.9439510n \\
&&
\end{array}\end{displaymath}

where n is an integer.




You can check each solution algebraically by substituting each solution in the original equation. If, after the substitution, the left side of the original equation equals the right side of the original equation, the solution is valid.


You can also check the solutions graphically by graphing the function formed by subtracting the right side of the original equation from the left side of the original equation. The solutions of the original equation are the x-intercepts of this graph.


Algebraic Check:


Check solution x=-0.9207554


Left Side:

\begin{displaymath}11\sin \left( \displaystyle \displaystyle \frac{3x}{10}\right...
...ystyle \frac{3\left( -0.9207554\right) }{10}\right) +4\approx 1\end{displaymath}

Right Side:        $1\bigskip $

Since the left side of the original equation equals the right side of the original equation when you substitute -0.9207554 for x, then -0.9207554is a solution.




Check solution x=11.39273095


Left Side:

\begin{displaymath}11\sin \left( \displaystyle \displaystyle \frac{3x}{10}\right...
...style \frac{3\left( 11.39273095\right) }{10}\right) +4\approx 1\end{displaymath}

Right Side:        $1\bigskip $

Since the left side of the original equation equals the right side of the original equation when you substitute 11.39273095 for x, then 11.39273095is a solution.




We have just verified algebraically that the exact solutions are $x=\displaystyle \displaystyle \frac{10%
}{3}\sin ^{-1}\left( -\displaystyle \displaystyle \frac{3}{11}\right) $ and $x=\displaystyle \frac{10}{3}\left( \pi
-\sin ^{-1}\left( -\displaystyle \displaystyle \frac{3}{11}\right) \right) $ and these solutions repeat every $\pm \displaystyle \displaystyle \frac{20\pi }{3}$ units. The approximate values of these solutions are $x\approx -0.9207554$ and $x\approx 11.39273095$ and these solutions repeat every $\pm 20.9439510$ units.




Graphical Check:


Graph the equation $f(x)=4\cos \left( \displaystyle \displaystyle \frac{x}{5}\right) +7-4$ (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. Let's check a few of these x-intercepts against the solutions we derived.


Verify the graph crosses the x-axis at -0.9207554. Since the period is $%
\displaystyle \displaystyle \frac{20\pi }{3}\approx 20.9439510$, you can verify that the graph also crosses the x-axis again at $-0.9207554+20.9439510\approx 20.0231956$ and at $-0.9207554+2\left( 20.9439510\right) =40.9671466$ etc.


Verify the graph crosses the x-axis at 11.39273095. Since the period is $%
\displaystyle \displaystyle \frac{20\pi }{3}\approx 20.9439510$, you can verify that the graph also crosses the x-axis again at $11.39273095+20.9439510\approx 32.336682$ and at $11.39273095+2\left( 20.9439510\right) =53.28063$ etc.


Note: If the problem were to find the solutions in the interval $\left[
0,2\pi \right] $, then you choose those solutions from the set of infinite solutions that belong to the set $\left[ 0,2\pi \right] .$ In this case, although there are an infinite number of solutions, none of the solutions are located in the interval $\left[ 0,2\pi \right] .\bigskip\bigskip
\bigskip\bigskip $



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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Author: Nancy Marcus

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