Question

Find the value of tan x if tan 2x=4/3

Answer 1

Answer:

$\tan \text{ x = -}\frac{1}{2}$

Explanation:

Here, we want to get the value of tan x

We start with the following trigonometric identities formula

We have this as:

$\tan \text{ 2x }=\text{ }\frac{2\text{ tan x}}{1-\tan ^2x}$

Let tan x = b

$\frac{4}{3}\text{ = }\frac{2b}{1-b^2}$

Now, let us solve for b

$\begin{gathered} 3(2b)=4(1-b^2) \\ 6b=4-4b^2 \\ 4b^2+6b-4\text{ = 0} \\ 2b^2-3b-2\text{ = 0} \end{gathered}$

We proceed to solve the quadratic equation as follows:

$\begin{gathered} 2b^2-4b+b-2\text{ = 0} \\ 2b(b-2)\text{ + 1(b-2) = 0} \\ (2b+1)(b-2)=0_{} \\ 2b\text{ + 1 = 0 or b-2 = 0} \\ b\text{ = -1/2 or b = 2} \\ \end{gathered}$

Thus, we have it that:

$\begin{gathered} \tan \text{ x = 2} \\ or\text{ } \\ \tan \text{ x = -1/2} \end{gathered}$

However, tan x = 2 will not be correct

This is because it will give a negative value of tan 2x

The only answer accepted is thus:

$\tan \text{ x = -1/2}$$\begin{gathered} \text{if tan x = 2} \\ \tan \text{ 2x = }\frac{2(2)}{1-(2)^2}\text{ = }\frac{4}{-3}\text{ = -}\frac{4}{3} \\ \\ \text{if tan x = -1/2} \\ \tan \text{ 2x =}\frac{2(-\frac{1}{2})}{1-(-\frac{1}{2})^2}\text{ = }\frac{-1}{1-\frac{1}{4}}\text{ =}\frac{-1}{-\frac{3}{4}}\text{ = }\frac{4}{3} \end{gathered}$