If sin `x=(sqrt(5))/(3) " and " 0 lt x lt .(pi)/(2)` find the values of
`(i) " sin " 2x " ""(ii) cos 2x "" ""(iii) tan 2x "`

Answer

To solve the problem step by step, let's follow the instructions given in the video transcript.

### Given:
- \( \sin x = \frac{\sqrt{5}}{3} \)
- \( 0 < x < \frac{\pi}{2} \)

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

Using the Pythagorean identity:
\[ \sin^2 x + \cos^2 x = 1 \]
Substituting the value of \( \sin x \):
\[ \left(\frac{\sqrt{5}}{3}\right)^2 + \cos^2 x = 1 \]
\[ \frac{5}{9} + \cos^2 x = 1 \]
\[ \cos^2 x = 1 - \frac{5}{9} = \frac{9}{9} - \frac{5}{9} = \frac{4}{9} \]
Taking the square root (since \( x \) is in the first quadrant where cosine is positive):
\[ \cos x = \frac{2}{3} \]

### Step 2: Find \( \sin 2x \)

Using the double angle formula:
\[ \sin 2x = 2 \sin x \cos x \]
Substituting the values of \( \sin x \) and \( \cos x \):
\[ \sin 2x = 2 \left(\frac{\sqrt{5}}{3}\right) \left(\frac{2}{3}\right) \]
\[ = \frac{4\sqrt{5}}{9} \]

### Step 3: Find \( \cos 2x \)

Using the double angle formula:
\[ \cos 2x = \cos^2 x - \sin^2 x \]
Substituting the values of \( \cos x \) and \( \sin x \):
\[ \cos 2x = \left(\frac{2}{3}\right)^2 - \left(\frac{\sqrt{5}}{3}\right)^2 \]
\[ = \frac{4}{9} - \frac{5}{9} \]
\[ = \frac{4 - 5}{9} = -\frac{1}{9} \]

### Step 4: Find \( \tan 2x \)

Using the double angle formula:
\[ \tan 2x = \frac{2 \tan x}{1 - \tan^2 x} \]
First, we need to find \( \tan x \):
\[ \tan x = \frac{\sin x}{\cos x} = \frac{\frac{\sqrt{5}}{3}}{\frac{2}{3}} = \frac{\sqrt{5}}{2} \]
Now substituting into the formula for \( \tan 2x \):
\[ \tan 2x = \frac{2 \left(\frac{\sqrt{5}}{2}\right)}{1 - \left(\frac{\sqrt{5}}{2}\right)^2} \]
Calculating \( \tan^2 x \):
\[ \tan^2 x = \left(\frac{\sqrt{5}}{2}\right)^2 = \frac{5}{4} \]
Now substituting:
\[ \tan 2x = \frac{2 \cdot \frac{\sqrt{5}}{2}}{1 - \frac{5}{4}} = \frac{\sqrt{5}}{1 - \frac{5}{4}} = \frac{\sqrt{5}}{\frac{-1}{4}} = -4\sqrt{5} \]

### Final Answers:
(i) \( \sin 2x = \frac{4\sqrt{5}}{9} \)
(ii) \( \cos 2x = -\frac{1}{9} \)
(iii) \( \tan 2x = -4\sqrt{5} \)