SOLVING TRIGONOMETRIC EQUATIONS


Note:

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


Solve for x in the following equation.


Example 3:        

$8\sin \left( \displaystyle \frac{1}{5}x\right) -3=0$


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

\begin{eqnarray*}&& \\ 8\sin \left( \displaystyle \frac{1}{5}x\right) -3 &=&0 \... ...e \frac{1}{5}x\right) &=&\displaystyle \frac{3}{8} \\ && \\ && \end{eqnarray*}


If we restriction the domain of the sine function to $\left[ -\displaystyle \frac{\pi }{2} \leq \displaystyle \frac{1}{5}x\leq ,\disp... ...\displaystyle \frac{ 5\pi }{2}\leq x\leq ,\displaystyle \frac{5\pi }{2}\right] $, we can use the inverse sine function to solve for reference angle $\displaystyle \frac{1}{5}x$ and then x.

\begin{eqnarray*}&& \\ \sin \left( \displaystyle \frac{1}{5}x\right) &=&\displa... ... &=&\sin ^{-1}\left( \displaystyle \frac{3}{8}\right) \\ && \\ \end{eqnarray*}
\begin{eqnarray*}\displaystyle \frac{1}{5}x &=&\sin ^{-1}\left( \displaystyle \f... ... x &\approx &1.92198387248\ \mbox{ radians }\\ && \\ && \\ && \end{eqnarray*}

We know that the $\sin $e function is positive in the first and the second quadrant. Therefore two of the solutions are the angle $\displaystyle \frac{1}{5}x$ that terminates in the first quadrant and the angle $\pi -\displaystyle \frac{1}{5}x$ that terminates in the second quadrant. We have already solved for $\displaystyle \frac{1}{5} x. $

\begin{eqnarray*}&& \\ \sin \left( \pi -\displaystyle \frac{1}{5}x\right) &=&\d... ...{5}x &=&\sin ^{-1}\left( \displaystyle \frac{3}{8}\right) \\ && \end{eqnarray*}
\begin{eqnarray*}&&\\ \displaystyle \frac{1}{5}x &=&\pi -\sin ^{-1}\left( \disp... ...&& \\ x &\approx &13.7859793955\ \mbox{ radians } \\ && \\ && \end{eqnarray*}

The exact solutions are $x=5\sin ^{-1}\left( \displaystyle \frac{3}{8}\right) $ and $ x=5\pi -5\sin ^{-1}\left( \displaystyle \frac{3}{8}\right) .\bigskip\bigskip $

The period of the sin $\left( \displaystyle \frac{1}{5}x\right) $ function is $10\pi .$This means that the values will repeat every $10\pi $ radians in both directions. Therefore, the exact solutions are $x=5\sin ^{-1}\left( \displaystyle \frac{3 }{8}\right) \pm n\left( 10\pi \right) $ and $x=5\pi -5\sin ^{-1}\left( \displaystyle \frac{ 3}{8}\right) \pm n\left( 10\pi \right) $ where n is an integer.


The approximate solutions are $x\approx 1.92198387248\pm n\left( 10\pi \right) $ and $x\approx 13.7859793955\pm n\left( 10\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=1.92198387248


Since the left side equals the right side when you substitute 1.92198387248for x, then 1.92198387248 is a solution.




Check the answer x=13.7859793955


Since the left side equals the right side when you substitute 13.7859793955for x, then 13.7859793955 is a solution.



Graphical Check:


Graph the equation

$f(x)=8\sin \left( \displaystyle \frac{1}{5}x\right) -3.\bigskip $

Note that the graph crosses the x-axis many times indicating many solutions. Note that it crosses at 1.92198387248.


Since the period is \begin{displaymath}10\pi \approx 31.415927 \end{displaymath} it crosses again at 1.92198387248 + 31.415927 = 33.3379104084 and at 1.92198387248 + 2( 31.415927 ) = 64.753838 , etc. The graph crosses at 13.7859793955 . Since the period is \begin{displaymath}10\pi \approx 31.415927 \end{displaymath} , it will cross again at 13.7859793955 + ( 31.415927 ) = 45.201906 and at 13.7859793955 + 2( 31.415927) = 76.617833, 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.


IF you would like to go to the next section, click on Next.


If you would like to go back to the equation table of contents, click on Contents.

This site was built to accommodate the needs of students. The topics and problems are what students ask for. We ask students to help in the editing so that future viewers will access a cleaner site. If you feel that some of the material in this section is ambiguous or needs more clarification, please let us know by e-mail.

[Algebra] [Trigonometry]
[Geometry] [Differential Equations]
[Calculus] [Complex Variables] [Matrix Algebra]

S.O.S MATHematics home page



Author: Nancy Marcus

Copyright © 1999-2024 MathMedics, LLC. All rights reserved.
Contact us
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA
users online during the last hour