SOLVING TRIGONOMETRIC EQUATIONS

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

Example 1:        Solve for x in the following equation.

$$3\tan ^{2}x-1=0$$

There are an infinite number of solutions to this problem. To solve for x, you must first isolate the tangent term.

$$\begin{array}{rclll} && \\ 3\tan ^{2}x-1 &=&0 \\ && \\ 3\t... ...&\pm \displaystyle \frac{1}{\sqrt{3}} \\ && \\ && \end{array}$$

If we restrict the domain of the tangent function to $\left( -\displaystyle \frac{\pi }{2}, \displaystyle \frac{\pi }{2}\right)$, we can use the inverse tangent function to solve for reference angle x', and then x.

$$\begin{array}{rclll} && \\ \tan x^{\prime } &=&\displaystyle... ... \\ x^{\prime } &\approx &0.52359877 \\ && \\ && \end{array}$$

The reference angle is $x^{\prime }=\tan ^{-1}\left( \displaystyle \displaystyle \frac{1}{\sqrt{3}} \right) .$ The tangent function is positive in the first quadrant and in the third quadrant and negative in the second and fourth quadrant.

The period of the $\tan \left( x\right)$ function is $\pi .$ This means that the values will repeat every $\pi$ radians in both directions. Therefore, the exact solutions are $x=\tan ^{-1}\left( \displaystyle \displaystyle \frac{1}{\sqrt{3}} \right) \pm n\left( \pi \right)$ and $x=\tan ^{-1}\left( -\displaystyle \frac{1}{\sqrt{3}}\right) \pm n\left( \pi \right)$ where n is an integer.

The approximate solutions are $x\approx 0.52359877\pm n\left( \pi \right)$and $x\approx -0.52359877\pm n\left( \pi \right)$ 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=0.52359877

Left Side:

$$3\tan ^{2}x-1\approx 3\tan ^{2}\left( 0.52359877\right) -1\approx 0\bigskip$$

Right Side:        0

Since the left side equals the right side when you substitute 0.52359877for x, then 0.52359877 is a solution.

Check answer . x=-0.52359877

Left Side:

$$3\tan ^{2}x-1\approx 3\tan ^{2}\left( -0.52359877\right) -1\approx 0\bigskip$$

Right Side:        0

Since the left side equals the right side when you substitute -0.52359877for x, then -0.52359877 is a solution.

Graphical Check:

Graph the equation $f(x)=3\tan ^{2}x-1.$ Note that the graph crosses the x-axis many times indicating many solutions.

Note that it crosses at 0.52359877. Since the period is $\pi \approx 3.1415927$, it crosses again at 0.52359877+3.1415927=3.66519 and at <tex2html_comment_mark> 0.52359877+2(3.1415927)=6.80678, etc.

Note that it crosses at -0.52359877. Since the period is $\pi \approx 3.1415927$, it crosses again at -0.52359877+3.1415927=2.617899 and at <tex2html_comment_mark> -0.52359877+2(3.1415927)=5.759587, 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.

If you would like to go to the next section, click on Next.

If you would like to go back to the equation table of contents, click on Contents.

[Algebra] [Trigonometry] [Geometry] [Differential Equations] [Calculus] [Complex Variables] [Matrix Algebra]

S.O.S MATHematics home page

Copyright © 1999-2004 MathMedics, LLC. All rights reserved.
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA