SOLVING TRIGONOMETRIC EQUATIONS

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

Example 1:        Solve for x in the following equation.

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

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

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

then

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

$\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

$$x\approx 1.5707963\pm 6.2831853n$$

and

$$x\approx 4.71238898\pm 6.2831853n$$

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:

$$\cot x\cos ^{2}x\approx \cot \left( 1.5707963\right) \cos ^{2}\left( 1.5707963\right) \approx 0$$

Right Side:

$$2\cot x\approx 2\cot \left( 1.5707963\right) \approx 0$$

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:

$$\cot x\cos ^{2}x\approx \cot \left( 4.71238898\right) \cos ^{2}\left( 4.71238898\right) \approx 0$$

Right Side:

$$2\cot x\approx 2\cot \left( 4.71238898\right) \approx 0$$

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 <tex2html_comment_mark> 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 <tex2html_comment_mark> 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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