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)=-√(3)\\ \\ 2x=(2\pi)/(3)+k\pi\\ \\ x=(\pi)/(3)+(k\pi)/(2) \end{gathered}`

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

`\begin{gathered} k=0:\\ \\ x=(\pi)/(3)\\ \\ \\ \\ k=1:\\ \\ x=(\pi)/(3)+(\pi)/(2)=(5\pi)/(6)\\ \\ \\ \\ k=2:\\ \\ x=(\pi)/(3)+\pi=(4\pi)/(3)\\ \\ \\ \\ k=3:\\ \\ x=(\pi)/(3)+(3\pi)/(2)=(11\pi)/(6) \end{gathered}`

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