If cos x = -1/3 and π < x < 3π/2, Find the value of cos x/2 ,tan x/2 . ​

Question

If cos x = \(\frac{-1}{3}\) and π < x < \(\frac{3\pi}{2}\) Find the value of cos \(\frac{x}{2}\),tan \(\frac{x}{2}\).

Answer 1

It is given that x lies in 3^rd quadrant.

∴ π < x < \(\frac{3\pi}{2}\)

⇒ \(\frac{\pi}{2}\)<\(\frac{\pi}{2}\)<\(\frac{3\pi}{4}\)

⇒ \(\frac{x}{2}\)lies in 2^nd quadrant.

⇒ cos x < 0, sin \(\frac{x}{2}\)> 0 and tan < 0

∴ cos\(\frac{x}{2}\) = ± \(\sqrt\frac{1+cos\,x}{2}\)

⇒  cos\(\frac{x}{2}\) = - \(\sqrt\frac{1+cos\,x}{2}\)

⇒  cos\(\frac{x}{2}\) = - \(\sqrt\frac{1-1/3}{2}\) = \(\sqrt\frac{2}{3}\)

∴ tan \(\frac{x}{2}\) = \(\frac{sin\,x/2}{cos\,x/2}\) = \(\frac{\sqrt\frac{2}{3}}{\frac{-1}{3}}\) = \(-\sqrt{2}\)