SOLVING TRIGONOMETRIC EQUATIONS

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

Problem 9.4b:        Solve for x in the following equation.

$$4\cot ^{2}x-3=0$$

Answers:        There are an infinite number of solutions: $x=\tan ^{-1}% \displaystyle \frac{2}{\sqrt{3}}\pm n\left( \pi \right)$ and $x=-\tan ^{-1}\displaystyle \displaystyle \frac{2}{% \sqrt{3}}\pm n\left( \pi \right)$ are the exact solutions, and $x\approx 0.8570719\pm n\left( \pi \right)$ and $x\approx -0.8570719\pm n\left( \pi \right)$ are the approximate solutions.

Solution:

To solve for x, first isolate the cotangent term.

$$\begin{array}{rclll} && \\ 4\cot ^{2}x-3 &=&0 \\ && \\ \co... ...an x &=&\pm \displaystyle \frac{2}{\sqrt{3}} \\ && \end{array}$$

If we restrict the domain of the tangent function to $-\displaystyle \displaystyle \frac{\pi }{2}<x<% \displaystyle \displaystyle \frac{\pi }{2}$, we can use the arctan function to solve for the reference angle $x^{\prime }$, and then x. The reference angle is always in the first quadrant.

$$\begin{array}{rclll} && \\ \tan x &=&\displaystyle \frac{2}{... ...rt{3}}\right) \\ && \\ x &\approx &0.85707194 \\ \end{array}$$

In the interval from $-\displaystyle \displaystyle \frac{\pi }{2}$ to $$\displaystyle \frac{\pi }{2}$$ the tangent of <tex2html_comment_mark> x is positive in the first quadrant and negative in the fourth quadrant.

One angle, $x_{1}=x^{\prime }=\tan ^{-1}\left( \displaystyle \displaystyle \frac{2}{\sqrt{3}}\right) \approx 0.85707194$ terminates in the first quadrant, and a fourth angle $% x_{4}=-x^{\prime }=-\tan ^{-1}\left( \displaystyle \displaystyle \frac{2}{\sqrt{3}}\right) \approx -0.85707194$ terminates in the fourth quadrant..

Since the period is $\pi ,$ this means that the rest of the solutions can be found by adding $\pi$ to each of the above solutions. Therefore, the solutions are

$$\begin{array}{rclll} && \\ x &=&\tan ^{-1}\left( \displaysty... ...rt{3}}\right) \pm n\left( \pi \right) \\ && \\ && \end{array}$$

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 the answer . x=0.85707194

Left Side:

$$4\cot ^{2}x-3\approx 4\cot ^{2}\left( 0.85707194\right) -3\approx 0\bigskip$$

Right Side:        $0\bigskip$

Since the left side equals the right side when you substitute 0.85707194for x, then 0.85707194 is a solution.

Check the answer . x=-0.85707194

Left Side:

$$4\cot ^{2}x-3\approx 4\cot ^{2}\left( -0.85707194\right) -3\approx 0\bigskip$$

Right Side:        $0\bigskip$

Since the left side equals the right side when you substitute -0.85707194for x, then -0.85707194 is a solution.

Graphical Check:

Graph the equation $f(x)=4\cot ^{2}x-3.$ (Formed by subtracting the right side of the original equation from the left side of the original equation.

Note that the graph crosses the x-axis many times indicating many solutions.

Note the graph crosses at -0.85707194 and -0.85707194. Since the period is $\pi \approx 3.1415926$, the graph will cross the x-axis again every $\pi$ units to the left and right of each number.

If you would like to review problem 9.4c, click on problem 9.4c.

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