Question

If sinx = √5/3 and π/2 < x < π, find the values of

(i) sin x/2

(ii) cos x/2

(iii) tan x/2

Answer 1

Given: sin x = \(\frac{\sqrt{5}}{3}\) and \(\frac{\pi}{2}\) < x < π i.e, x lies in the Quadrant II .

To Find: 

(i) sin\(\frac{x}{2}\)

(ii) cos\(\frac{x}{2}\)

(iii) tan\(\frac{x}{2}\)

Now, since sin x = \(\frac{\sqrt{5}}{3}\)

We know that cos x = \(\pm \sqrt{1-sin^2\text{x}}\)

since cos x is negative in II quadrant, hence cos x = \(-\frac{2}{3}\)

(i) sin\(\frac{x}{2}\)

Formula used:

Since sinx is positive in II quadrant, hence sin \(\frac{x}{2}=\sqrt{\frac{5}{6}}{}\)

(ii) cos\(\frac{x}{2}\)

Formula used:

since cosx is negative in II quadrant, hence cos\(\frac{x}{2}=-\frac{1}{\sqrt{6}}\)

(iii) tan\(\frac{x}{2}\)

Formula used:

Here, tan x is negative in II quadrant.