If sin `x=(sqrt(5))/(3) "and " .(pi)/(3) lt x lt pi ` find the values of
`(i) " sin " (x)/(2) " " (ii) " cos " (x)/(2) " " (iii) " tan " (x)/(2)`

Answer

To solve the problem, we need to find the values of \( \sin \frac{x}{2} \), \( \cos \frac{x}{2} \), and \( \tan \frac{x}{2} \) given that \( \sin x = \frac{\sqrt{5}}{3} \) and \( \frac{\pi}{3} < x < \pi \).

### Step 1: Find \( \cos x \)

We know that \( \sin^2 x + \cos^2 x = 1 \).

Given \( \sin x = \frac{\sqrt{5}}{3} \):

\[ \sin^2 x = \left(\frac{\sqrt{5}}{3}\right)^2 = \frac{5}{9} \]

Now, substituting into the Pythagorean identity:

\[ \cos^2 x = 1 - \sin^2 x = 1 - \frac{5}{9} = \frac{4}{9} \]

Thus,

\[ \cos x = \pm \sqrt{\frac{4}{9}} = \pm \frac{2}{3} \]

Since \( x \) is in the interval \( \left(\frac{\pi}{3}, \pi\right) \), where cosine is negative, we have:

\[ \cos x = -\frac{2}{3} \]

### Step 2: Find \( \sin \frac{x}{2} \) using the half-angle formula

The half-angle formula for sine is:

\[ \sin \frac{x}{2} = \sqrt{\frac{1 - \cos x}{2}} \]

Substituting \( \cos x = -\frac{2}{3} \):

\[ \sin \frac{x}{2} = \sqrt{\frac{1 - \left(-\frac{2}{3}\right)}{2}} = \sqrt{\frac{1 + \frac{2}{3}}{2}} = \sqrt{\frac{\frac{5}{3}}{2}} = \sqrt{\frac{5}{6}} \]

### Step 3: Find \( \cos \frac{x}{2} \) using the half-angle formula

The half-angle formula for cosine is:

\[ \cos \frac{x}{2} = \sqrt{\frac{1 + \cos x}{2}} \]

Substituting \( \cos x = -\frac{2}{3} \):

\[ \cos \frac{x}{2} = \sqrt{\frac{1 + \left(-\frac{2}{3}\right)}{2}} = \sqrt{\frac{1 - \frac{2}{3}}{2}} = \sqrt{\frac{\frac{1}{3}}{2}} = \sqrt{\frac{1}{6}} \]

### Step 4: Find \( \tan \frac{x}{2} \)

Using the relationship \( \tan \frac{x}{2} = \frac{\sin \frac{x}{2}}{\cos \frac{x}{2}} \):

\[ \tan \frac{x}{2} = \frac{\sqrt{\frac{5}{6}}}{\sqrt{\frac{1}{6}}} = \sqrt{5} \]

### Final Answers

(i) \( \sin \frac{x}{2} = \sqrt{\frac{5}{6}} \)
(ii) \( \cos \frac{x}{2} = \sqrt{\frac{1}{6}} \)
(iii) \( \tan \frac{x}{2} = \sqrt{5} \)