1. Given; sin x = \(\frac{\sqrt{3}}{2}\) = sin \(\frac{\pi}{3}\)

General solution is; x = nπ + (-1)ⁿ \(\frac{\pi}{3}\)

Put n = 0, 1 we get principal solution; x = \(\frac{\pi}{3}\); \(\frac{2\pi}{3}\)

2. Given; cos x = \(\frac{1}{2}\)= cos \(\frac{\pi}{3}\) 

General solution is; x = 2nπ ± \(\frac{\pi}{3}\), n ∈ Z 

Put n = 0, 1 we get principal solution; 

n = 0 

⇒ x = \(\frac{\pi}{3}\) ; n = 1 

⇒ x = 2π – \(\frac{\pi}{3}\) = \(\frac{5 \pi}{3}.\)

3. Given; tan x = \(\sqrt{3}\) = tan \(\frac{\pi}{3}\) 

General solution is; ⇒ x = nπ + \(\frac{\pi}{3}\), n ∈ Z 

Put n = 0, 1 we get principal solution;

n = 0 \(\Rightarrow\) \(\frac{\pi}{3}\); n = 1 \(\Rightarrow\) x = π + \(\frac{\pi}{3}\) = 4\(\frac{\pi}{3}\).

4. Given; cosec x = -2 

⇒ sin x = \(\frac{-1}{2}\) = – sin \(\frac{\pi}{6}\)= sin(-\(\frac{\pi}{6}\)) 

General solution is; x = nπ – (-1) , n ∈ Z 

Put n = 1, 2 we get principal solution;

⇒ n = 1 ⇒ x = \(\pi + \frac{\pi}{6} = \frac{7\pi}{6}\)

⇒ n = 2 ⇒ x = 2\(\pi + \frac{\pi}{6} = \frac{11\pi}{12}\)