SOLVING TRIGONOMETRIC EQUATIONS


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


Example 2:        Solve for x in the following equation.



\begin{displaymath}4\cos ^{2}x-3=0\end{displaymath}

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


\begin{displaymath}\begin{array}{rclll}
&& \\
4\cos ^{2}x-3 &=&0 \\
&& \\
4\c...
...tyle \displaystyle \frac{\sqrt{3}}{2} \\
&& \\
&&
\end{array}\end{displaymath}

If we restrict the domain of the cosine function to $\left[ 0\leq x\leq \pi
\right] $, we can use the inverse cosine function to solve for reference angle $x^{\prime }$, and then x. The reference angle is always located in the first quadrant and positive.

\begin{displaymath}\begin{array}{rclll}
&& \\
\cos x &=&\displaystyle \displays...
...{2}\right) \\
&& \\
x &\approx &0.52359877 \\
&&
\end{array}\end{displaymath}

The cosine function is positive in the first and fourth quadrants and negative in the second and third quadrants. We will use the reference angle $%
x^{\prime }$ to find the four angles.



The first solution is the angle that terminates in the first quadrant: is $%
x_{1}=x^{\prime }= \cos ^{-1}\left( \displaystyle \displaystyle \frac{\sqrt{3}}{2}\right) .$ The second solution is the angle that terminates in the second quadrant: $x_{2}=\pi
-x^{\prime }=\pi - \cos ^{-1}\left( \displaystyle \displaystyle \frac{\sqrt{3}}{2}\right) $. The third solution is the angle that terminates in the third quadrant: $x_{3}=\pi
+ \cos ^{-1}\left( \displaystyle \displaystyle \frac{\sqrt{3}}{2}\right) .$ The fourth solutions is the angle that terminates in the fourth quadrant: $2\pi - \cos ^{-1}\left( \displaystyle \displaystyle \frac{%
\sqrt{3}}{2}\right) .\bigskip\bigskip $

The period of the cos $\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

\begin{displaymath}\begin{array}{rclll}
&& \\
x &=& \cos ^{-1}\left( \displayst...
...qrt{3}}{2}\right) \pm n\left( 2\pi
\right) , \\
&&
\end{array}\end{displaymath}

where n is an integer.


The approximate solutions are $x\approx 0.52359877\pm n\left( 2\pi \right) ,$ $x\approx 2.6179938\pm n\left( 2\pi \right) $, $x\approx 3.66519\pm n\left(
2\pi \right) $ and $x\approx 5.759586537\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=0.52359877


Left Side:

\begin{displaymath}4\cos ^{2}x-3\approx 4\cos ^{2}\left( 0.52359877\right)
-3\approx 0\bigskip\end{displaymath}

Right Side:        $0\bigskip $

Since the left side equals the right side when you substitute 0.52359877for x, then 0.52359877 is a solution.




Check answer . x=2.6179938


Left Side:

\begin{displaymath}4\cos ^{2}x-3\approx 4\cos ^{2}\left( 2.6179938\right)
-3\approx 0\bigskip\end{displaymath}

Right Side:        $0\bigskip $

Since the left side equals the right side when you substitute 2.6179938for x, then 2.6179938 is a solution.




Check answer .x=3.66519


Left Side:

\begin{displaymath}4\cos ^{2}x-3\approx 4\cos ^{2}\left( 3.66519\right)
-3\approx 0\bigskip\end{displaymath}

Right Side:        $0\bigskip $

Since the left side equals the right side when you substitute 3.66519 for x, then 3.66519 is a solution.




Check answer . x=5.759586537


Left Side:

\begin{displaymath}4\cos ^{2}x-3\approx 4\cos ^{2}\left( 5.759586537\right)
-3\approx 0\bigskip\end{displaymath}

Right Side:        $0\bigskip $

Since the left side equals the right side when you substitute 5.759586537for x, then 5.759586537 is a solution.



Graphical Check:


Graph the equation $4\cos ^{2}x-3.$ Note that the graph crosses the x-axis many times indicating many solutions.


Note that it crosses four time in the interval from 0 ro $2\pi
:0.52359877,\ 2.6179938$, 3.66519 and 5.759586537.


Since the period is $2\pi $, the graph crosses again at $0.52359877+2\pi ,\
2.617899$ $+2\pi $, $3.66519+2\pi $, and $5.759586537+2\pi $ 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.


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[Geometry] [Differential Equations]
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Author: Nancy Marcus

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