SOLVING TRIGONOMETRIC EQUATIONS


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


Example 1:        Solve for x in the following equation.



\begin{displaymath}3\tan ^{2}x-1=0\end{displaymath}

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


\begin{displaymath}\begin{array}{rclll}
&& \\
3\tan ^{2}x-1 &=&0 \\
&& \\
3\t...
...&\pm \displaystyle \frac{1}{\sqrt{3}} \\
&& \\
&&
\end{array}\end{displaymath}

If we restrict the domain of the tangent function to $\left( -\displaystyle \frac{\pi }{2}, \displaystyle \frac{\pi }{2}\right)$, we can use the inverse tangent function to solve for reference angle x', and then x.

\begin{displaymath}\begin{array}{rclll}
&& \\
\tan x^{\prime } &=&\displaystyle...
... \\
x^{\prime } &\approx &0.52359877 \\
&& \\
&&
\end{array}\end{displaymath}

The reference angle is $x^{\prime }=\tan ^{-1}\left( \displaystyle \displaystyle \frac{1}{\sqrt{3}}
\right) .$ The tangent function is positive in the first quadrant and in the third quadrant and negative in the second and fourth quadrant.



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

The approximate solutions are $x\approx 0.52359877\pm n\left( \pi \right) $and $x\approx -0.52359877\pm n\left( \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}3\tan ^{2}x-1\approx 3\tan ^{2}\left( 0.52359877\right)
-1\approx 0\bigskip\end{displaymath}

Right Side:        0

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




Check answer . x=-0.52359877


Left Side:

\begin{displaymath}3\tan ^{2}x-1\approx 3\tan ^{2}\left( -0.52359877\right)
-1\approx 0\bigskip\end{displaymath}

Right Side:        0

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



Graphical Check:


Graph the equation $f(x)=3\tan ^{2}x-1.$ Note that the graph crosses the x-axis many times indicating many solutions.


Note that it crosses at 0.52359877. Since the period is $\pi \approx
3.1415927$, it crosses again at 0.52359877+3.1415927=3.66519 and at <tex2htmlcommentmark> 0.52359877+2(3.1415927)=6.80678, etc.


Note that it crosses at -0.52359877. Since the period is $\pi \approx
3.1415927$, it crosses again at -0.52359877+3.1415927=2.617899 and at <tex2htmlcommentmark> -0.52359877+2(3.1415927)=5.759587, 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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Author: Nancy Marcus

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