SOLVING TRIGONOMETRIC EQUATIONS

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

Problem 9.5a: Solve for x in the equation

$$2\tan x\cos ^{2}x=\tan x$$

Answer:    The exact answers are

$$\begin{array}{rclll} x_{1} &=&0\pm n\pi \\ && \\ x_{2} &=&\... ...isplaystyle \frac{1}{\sqrt{2}}\right) \pm 2n\pi \\ \end{array}$$

where n is an integer.

The approximate values of these solutions are

$$\begin{array}{rclll} x_{1} &\approx &0\pm 3.141592653n \\ x_... ...\ x_{6} &\approx &-2.35619449\pm 6.2831853n \\ && \end{array}$$

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.

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

then

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

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.

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

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

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

The exact solutions are

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

where n is an integer.

The approximate values of these solutions are

$$\begin{array}{rclll} x_{1} &\approx &\pm 6.2831853n \\ x_{2}... ...n \\ x_{6} &\approx &-2.35619447\pm 6.2831853n \\ \end{array}$$

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:

$$2\tan x\cos ^{2}x\approx 2 \tan\left( 0\right) \cos ^{2}\left( 0\right) \approx 0$$

Right Side:

$$\tan x\approx \tan \left( 0\right) \approx 0$$

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:

$$2\tan x\cos ^{2}x\approx 2 \tan\left( 3.141592653\right) \cos ^{2}\left( 3.141592653\right) \approx 0$$

Right Side:

$$\tan x\approx \tan \left( 3.141592653\right) \approx 0$$

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:

$$2\tan x\cos ^{2}x\approx 2 \tan\left( 0.785398\right) \cos ^{2}\left( 0.785398\right) \approx 1$$

Right Side:

$$\tan x\approx \tan \left( 0.785398\right) \approx 1$$

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:

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

Right Side:

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

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:

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

Right Side:

$$\tan x\approx \tan \left( -0.785398\right) \approx 1$$

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:

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

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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