Question

Find the principal and general solutions of the equation $\tan x=\sqrt{3}$

Answer 1

Step $1:$ Solve for principal solution.
Given that $\tan x=\sqrt{3}$

We know that $\tan {60}^{\circ }=\sqrt{3}$
Here $\tan x$ is positive.

$\tan$ is positive in $1^{st}$ and $3^{rd}$ quadrant
Value in $1^{st}$ quadrant $={60}^{\circ }$
Value in $3^{rd}$ quadrant $={180}^{\circ }+{60}^{\circ }={240}^{\circ }$

So principal solutons are $x={60}^{\circ }=60\times \frac{\pi }{180}=\frac{\pi }{3}$ and $x={240}^{\circ }=240\times \frac{\pi }{180}=\frac{4\pi }{3}$

Step 2 : Solve for general solution.
$\tan x=\sqrt{3}$
$\Rightarrow \tan x=\tan \frac{\pi }{3}$

We know that if $\tan x=\tan y$
General solution is $x=n\pi +y$ where $n\in Z$
Put $y=\frac{\pi }{3}$
Hence, $x=n\pi +\frac{\pi }{3},$ where $n\in Z$