SOLVING TRIGONOMETRIC EQUATIONS

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Problem 9.6d: Solve for x in the equation

$$3\tan ^{2}x+25\tan x-50=0$$

Answer:     The exact answers are

$$\begin{array}{rclll} x_{1} &=&\tan ^{-1}\left( \displaystyle ... ... x_{2} &=&\tan ^{-1}\left( -10\right) \pm n\pi \\ \end{array}$$

where n is an integer.

The approximate values of these solutions are

$$\begin{array}{rclll} x_{1} &\approx &1.030377\pm 3.141593n \\ && \\ x_{2} &\approx &-1.47112767\pm 3.141593n \\ \end{array}$$

where n is an integer.

Solution:

There are an infinite number of solutions to this problem.

Isolate the $\tan x$. To do this rewrite the left side of the equation in an equivalent factored form.

$$\begin{array}{rclll} 3\tan ^{2}x+25\tan x-50 &=&0 \\ && \\ ... ...( 3\tan x-5\right) \left( \tan x+10\right) &=&0 \\ \end{array}$$

The only way the product of two factors equals zero is if at least one of the factors equals zeros. This means that $3\tan ^{2}x+25\tan x-50=0$ if $3\tan x-5=0\$or $\tan x+10=0.$

We have transformed a difficult problem into two easier problems. To find the solutions to the original equation, $\left( 3\tan x-5\right) \left( \tan x+10\right) =0$, just find the solutions to the equations $3\tan x-5=0\$ and $\tan x+10=0.$

$$\begin{array}{rclll} 3\tan x-5 &=&0 \\ && \\ \tan x &=&\displaystyle \frac{5}{3} \\ \end{array}$$

and

$$\begin{array}{rclll} \tan x+10 &=&0 \\ && \\ \tan x &=&-10 \\ \end{array}$$

How do we isolate the x? We could take the arctangent of both sides. However, the tangent function is not a one-to-one function.

We can restrict the domain of the function so that the function is one-to-one on the restricted domain while preserving the original range. The graph of the tangent function is one-to-one on the interval $\left( -\displaystyle \displaystyle \frac{\pi }{2}\displaystyle \displaystyle \frac{\pi }{2}\right) .$ If we restrict the domain of the tangent function to that interval , we can take the arctangent of both sides of each equation.

$$\begin{array}{rclll} (1)\qquad \tan x &=&\displaystyle \frac{... ...isplaystyle \frac{5}{3}\right) \approx 1.030377 \\ \end{array}$$

and

$$\begin{array}{rclll} (2)\qquad \tan x &=&-10 \\ && \\ \tan ... ...&\tan ^{-1}\left( -10\right) \approx -1.4711277 \\ \end{array}$$

Since the period of $\tan x$ equals $\pi$, these solutions will repeat every $\pi$ units. The exact solutions are

$$\begin{array}{rclll} x_{1} &=&\tan ^{-1}\left( \displaystyle ... ... x_{2} &=&\tan ^{-1}\left( -10\right) \pm n\pi \\ \end{array}$$

where n is an integer.

The approximate values of these solutions are

$$\begin{array}{rclll} x_{1} &\approx &1.030377\pm 3.14159265n ... ...& \\ x_{2} &\approx &-1.4711277\pm 3.14159265n \\ \end{array}$$

where n is an integer.

You can check each solution algebraically by substituting each solution in the original equation. If, after the substitution, the left side of the original equation equals the right side of the original equation, the solution is valid.

You can also check the solutions graphically by graphing the function formed by the left side of the original equation and graphing the function formed by the right side of the original equation. The x-coordinates of the points of intersection are the solutions. The right side of the equation is 0 and f(x)=0 is the x-axis. So really what you are looking for are the x-intercepts to the function formed by the left side of the equation.

Algebraic Check:

Check solution $x=\tan ^{-1}\left( \displaystyle \displaystyle \frac{5}{3}\right) \approx 1.030377$

Left Side:

$$3\tan ^{2}x+25\tan x-50=0\approx 3\tan ^{2}\left( 1.030377\right) +25\tan \left( 1.030377\right) -50=0\approx 0$$

Right Side:        0

Since the left side of the original equation equals the right side of the original equation when you substitute 1.030377 for x, then 1.030377 is a solution.

Check solution $x=\tan ^{-1}\left( \displaystyle \displaystyle \frac{5}{3}\right) \approx -1.4711277$

Left Side:

$$3\tan ^{2}x+25\tan x-50=0\approx 3\tan ^{2}\left( -1.4711277\right) +25\tan \left( -1.4711277\right) -50=0\approx 0$$

Right Side:        0

Since the left side of the original equation equals the right side of the original equation when you substitute -1.4711277 for x, then -1.4711277 is a solution.

Graphical Check:

Graph the equation $f(x)=3\tan ^{2}x+25\tan x-50.$ Note that the graph crosses the x-axis many times indicating many solutions.

You can verify that the graph crosses the x-axis at 1.030377. Since the period is $\pi \approx 3.1415927$, the graph also crosses the x-axis again at 1.030377+3.1415927=4.1719697 and at $1.030377+2\left( 3.1415927\right) =7.31356$, etc.

You can verify that the graph crosses the x-axis at -1.4711277. Since the period is $\pi \approx 3.1415927$, the graph also crosses the x-axis again at -1.4711277+3.1415927=1.670465 and at $-1.4711277+2\left( 3.1415927\right) =4.8120577$, etc.

Note: If the problem was to find the solutions in the interval $\left[ 0,2\pi \right]$, then you choose those solutions from the set of infinite solutions that are belong to the set $\left[ 0,2\pi \right] :$ $x\approx 1.030377,$ 4.1719697, 1.670465 and 4.8120577.

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