SOLVING TRIGONOMETRIC EQUATIONS

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


Problem 9.5a: Solve for x in the equation

\begin{displaymath}2\tan x\cos ^{2}x=\tan x\end{displaymath}

Answer:    The exact answers are

\begin{displaymath}\begin{array}{rclll}
x_{1} &=&0\pm n\pi \\
&& \\
x_{2} &=&\...
...isplaystyle \frac{1}{\sqrt{2}}\right) \pm 2n\pi \\
\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\pm 3.141592653n \\
x_...
...\
x_{6} &\approx &-2.35619449\pm 6.2831853n \\
&&
\end{array}\end{displaymath}

where n is an integer.

Solution:

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}
2\tan x\cos ^{2}x &=&\tan x \\
&& \\
2...
...
&& \\
\tan x\left( 2\cos ^{2}x-1\right) &=&0 \\
\end{array}\end{displaymath}

then

\begin{displaymath}\begin{array}{rclll}
\tan x &=&0 \\
or&& \\
2\cos ^{2}x-1 &=&0 \\
\end{array}\end{displaymath}

The function $\tan x=\displaystyle \displaystyle \frac{\sin x}{\cos x}=0$ when $\sin x=0$ and $\sin x=0$ when x=0 or $\pi$.

$2\cos ^{2}x-1=0$ when $\cos ^{2}x=\displaystyle \frac{1}{2}$ and $\cos x=\pm \sqrt{\displaystyle \frac{1}{2}}.$

How do we isolate the x in the equations $\cos x=\pm \sqrt{\displaystyle \displaystyle \frac{1}{2}}$? We could take the arccosine of both sides. However, the cosine function is not a one-to-one function.

Let's restrict the domain so the function is one-to-one on the restricted domain while preserving the original range. The graph of the cosine function is one-to-one on the interval $\left[ 0,\pi \right] .$ If we restrict the domain of the cosine function to that interval , we can take the arccosine of both sides of each equation.

\begin{displaymath}\begin{array}{rclll}
\cos x &=&\sqrt{\displaystyle \frac{1}{2...
...laystyle \frac{1}{2}}\right) \approx 2.35619449 \\
\end{array}\end{displaymath}

Since $\cos \left( -x\right) =\cos \left( x\right) ,$ we know that if $\cos
x=\pm \sqrt{\displaystyle \frac{1}{2}},$ then $\cos \left( -x\right) =\pm \sqrt{\displaystyle \frac{1}{2}}$ and

\begin{displaymath}\begin{array}{rclll}
x &=&-\cos ^{-1}\left( \sqrt{\displaysty...
...tyle \frac{1}{2}}\right) \approx -2.35619449 \\
&&
\end{array}\end{displaymath}

The exact solutions are

\begin{displaymath}\begin{array}{rclll}
x_{1} &=&0\pm 2n\pi \\
&& \\
x_{2} &=&...
...ft( -\displaystyle \frac{1}{\sqrt{2}}\right) \\
&&
\end{array}\end{displaymath}

where n is an integer.

The approximate values of these solutions are

\begin{displaymath}\begin{array}{rclll}
x_{1} &\approx &\pm 6.2831853n \\
x_{2}...
...n \\
x_{6} &\approx &-2.35619447\pm 6.2831853n \\
\end{array}\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 answer x=0

Left Side:

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

Right Side:

\begin{displaymath}\tan x\approx \tan \left( 0\right) \approx 0 \end{displaymath}

Since the left side equals the right side when you substitute 0 for x, then 0 is a solution.

Check answer x=3.141592653

Left Side:

\begin{displaymath}2\tan x\cos ^{2}x\approx 2 \tan\left( 3.141592653\right) \cos
^{2}\left( 3.141592653\right) \approx 0 \end{displaymath}

Right Side:

\begin{displaymath}\tan x\approx \tan \left( 3.141592653\right) \approx
0 \end{displaymath}

Since the left side equals the right side when you substitute 3.141592653 for x, then 3.141592653 is a solution.

Check answer x=0.785398

Left Side:

\begin{displaymath}2\tan x\cos ^{2}x\approx 2 \tan\left( 0.785398\right) \cos
^{2}\left( 0.785398\right) \approx 1 \end{displaymath}

Right Side:

\begin{displaymath}\tan x\approx \tan \left( 0.785398\right) \approx 1 \end{displaymath}

Since the left side equals the right side when you substitute 0.785398 for x, then 0.785398 is a solution.

Check answer x=2.35619447

Left Side:

\begin{displaymath}2\tan x\cos ^{2}x\approx 2 \tan\left( 2.35619447\right) \cos^{2}\left( 2.35619447\right) \approx -1 \end{displaymath}

Right Side:

\begin{displaymath}\tan x\approx \tan \left( 2.35619447\right) \approx -1 \end{displaymath}

Since the left side equals the right side when you substitute 2.35619447for x, then 2.35619447 is a solution.

Check answer x=-0.785398

Left Side:

\begin{displaymath}2\tan x\cos ^{2}x\approx 2 \tan\left( -0.785398\right) \cos
^{2}\left( -0.785398\right) \approx 1 \end{displaymath}

Right Side:

\begin{displaymath}\tan x\approx \tan \left( -0.785398\right) \approx 1 \end{displaymath}

Since the left side equals the right side when you substitute -0.785398for x, then -0.785398 is a solution.

Check answer x=-2.35619447

Left Side:

\begin{displaymath}2\tan x\cos ^{2}x\approx 2 \tan\left( -2.35619447\right) \cos
^{2}\left( -2.35619447\right) \approx -1 \end{displaymath}

Right Side:         $\tan x\approx \tan \left( -2.35619447\right) \approx
-1 $

Since the left side equals the right side when you substitute -2.35619447 for x, then -2.35619447 is a solution.

Graphical Check:

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

Verify that it crosses at 0. Since the period is $2\pi \approx 6.2831853$, it crosses again at 0+6.2831853=6.2831853 and at 0+2(6.2831853)=12.5663706, etc.

Verify that it crosses at 3.141592653. Since the period is $2\pi \approx 6.2831853$, it crosses again at 3.141592653+6.2831853=9.42477796 and at 3.141592653+2(6.2831853)=15.7079633, etc.

Verify that it crosses at 0.785398. Since the period is $2\pi \approx 6.2831853$, it crosses again at 0.785398+6.2831853=7.0685833 and at 0.785398+2(6.2831853)=13.3517686, etc.

Verify that it crosses at 2.35619447. Since the period is $2\pi \approx 6.2831853$, it crosses again at 2.35619447+6.2831853 = 78.639379777 and at 2.35619447+2(6.2831853)=14.922565, etc.

Verify that it crosses at -0.785398. Since the period is $2\pi \approx 6.2831853$, it crosses again at -0.785398+6.2831853=5.4977873 and at -0.785398+2(6.2831853)=11.7809726, etc.

Verify that it crosses at -2.35619447. Since the period is $2\pi \approx 6.2831853$, it crosses again at -2.35619447+6.2831853=3.9269908 and at -2.35619447+2(6.2831853)=10.210176, etc.



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Author: Nancy Marcus

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