Hello everyone!

I've found the answer to this just need to know if its in the right term

Given that cosec θ = −2, tan θ = 1/3√3

and −π < θ < π, find the exact

value of the angle θ in radians. Justify your answer.

answers are

cosec = 1/sin = -1/2 (which means its in the 3rd or 4th quadrant)

tan = 1/3√3 (which means its in the 1st or 3rd quadrant)

this means that theta is in the 3rd quadrant

but is it -1/6pi or -5/6pi? because each of their values is either +/- 1/6pi but surely that cant be right because -1/6pi is -30 which is not in the 3rd quadrant, could someone just classify this for me! cheers