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}\cot x\cos ^{2}x=2\cot x\end{displaymath}

There are an infinite number of solutions to this problem. To solve for x, set the equation equal to zero and factor.


\begin{displaymath}\begin{array}{rclll}
\cot x\cos ^{2}x &=&2\cot x \\
&& \\
\...
...\
&& \\
\cot x\left( \cos ^{2}x-2\right) &=&0 \\
\end{array}\end{displaymath}

then

\begin{displaymath}\begin{array}{rclll}
\cot x &=& 0 \\
or&& \\
\cos ^{2}x-2 &=&0 \\
\end{array}\end{displaymath}

$\cot x=\displaystyle \frac{\cos x}{\sin x}=0\ $when $\cos x=0.$ $\cos x=0$ when $x=\displaystyle \frac{\pi }{2}\pm 2\pi $, and when $x=\displaystyle \frac{3\pi }{2}\pm 2\pi .$

$\cos ^{2}x-2=0$ when $\cos ^{2}x=2$ and $\cos x=\pm \sqrt{2}.$ This is impossible because $-1\leq \cos x\leq 1. $

The exact value solutions are $x=\displaystyle \frac{\pi }{2}\pm 2\pi $ and $x=\displaystyle \frac{%
3\pi }{2}\pm 2\pi .$ The approximate value of these solutions are

\begin{displaymath}x\approx 1.5707963\pm 6.2831853n\end{displaymath}

and

\begin{displaymath}x\approx 4.71238898\pm 6.2831853n\end{displaymath}

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=1.5707963

Left Side:

\begin{displaymath}\cot x\cos ^{2}x\approx \cot \left( 1.5707963\right) \cos
^{2}\left( 1.5707963\right) \approx 0 \end{displaymath}

Right Side:

\begin{displaymath}2\cot x\approx 2\cot \left( 1.5707963\right) \approx
0\end{displaymath}

Since the left side equals the right side when you substitute 1.5707963for x, then 1.5707963 is a solution.

Check the answer x=4.71238898

Left Side:

\begin{displaymath}\cot x\cos ^{2}x\approx \cot \left( 4.71238898\right)
\cos ^{2}\left( 4.71238898\right) \approx 0 \end{displaymath}

Right Side:

\begin{displaymath}2\cot x\approx 2\cot \left( 4.71238898\right) \approx
0\end{displaymath}

Since the left side equals the right side when you substitute 4.71238898for x, then 4.71238898 is a solution.

Graphical Check:

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

Note that it crosses at 1.5707963. Since the period is $2\pi \approx
6.2831853$, it crosses again at 1.5707963+6.2831853=7.85398 and at <tex2htmlcommentmark> 1.5707963+2(6.2831853)=14.137167, etc.

Note that it crosses at 4.71238898. Since the period is $2\pi \approx
6.2831853$, it crosses again at 4.71238898+6.2831853=10.99557 and at <tex2htmlcommentmark> 4.71238898+2(6.2831853)=17.27876, 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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