Question

find all solutions for x in [0,2TT)tan(2x) = - root 3

Answer 1

In order to find the solutions, let's solve the equation for x:

$\begin{gathered} \tan(2x)=-\sqrt{3}\\ \\ 2x=\frac{2\pi}{3}+k\pi\\ \\ x=\frac{\pi}{3}+\frac{k\pi}{2} \end{gathered}$

Using l = 0, k = 1, k = 2 and k = 3, we have:

$\begin{gathered} k=0:\\ \\ x=\frac{\pi}{3}\\ \\ \\ \\ k=1:\\ \\ x=\frac{\pi}{3}+\frac{\pi}{2}=\frac{5\pi}{6}\\ \\ \\ \\ k=2:\\ \\ x=\frac{\pi}{3}+\pi=\frac{4\pi}{3}\\ \\ \\ \\ k=3:\\ \\ x=\frac{\pi}{3}+\frac{3\pi}{2}=\frac{11\pi}{6} \end{gathered}$

Therefore the solutions are pi/3, 5pi/6, 4pi/3 and 11pi/6.