If `tan x = -(4)/(3), (pi)/(2) lt x lt pi`, then find the value of `sin(x/2), cos(x/2)` and `tan(x/2)`.

Answer

To solve the problem, we need to find the values of \( \sin\left(\frac{x}{2}\right) \), \( \cos\left(\frac{x}{2}\right) \), and \( \tan\left(\frac{x}{2}\right) \) given that \( \tan x = -\frac{4}{3} \) and \( \frac{\pi}{2} < x < \pi \).

### Step 1: Determine the value of \( \sin x \) and \( \cos x \)

Since \( \tan x = \frac{\sin x}{\cos x} \), we can use the relationship:

\[ \tan x = -\frac{4}{3} \implies \sin x = -4k \quad \text{and} \quad \cos x = 3k \]

where \( k \) is a positive constant.

Using the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \):

\[ (-4k)^2 + (3k)^2 = 1 \]

\[ 16k^2 + 9k^2 = 1 \]

\[ 25k^2 = 1 \implies k^2 = \frac{1}{25} \implies k = \frac{1}{5} \]

Now substituting back to find \( \sin x \) and \( \cos x \):

\[ \sin x = -4k = -4 \cdot \frac{1}{5} = -\frac{4}{5} \]

\[ \cos x = 3k = 3 \cdot \frac{1}{5} = \frac{3}{5} \]

### Step 2: Find \( \sin\left(\frac{x}{2}\right) \) and \( \cos\left(\frac{x}{2}\right) \)

Using the half-angle formulas:

\[ \sin\left(\frac{x}{2}\right) = \sqrt{\frac{1 - \cos x}{2}} \quad \text{and} \quad \cos\left(\frac{x}{2}\right) = \sqrt{\frac{1 + \cos x}{2}} \]

Substituting \( \cos x = \frac{3}{5} \):

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

\[ \cos\left(\frac{x}{2}\right) = \sqrt{\frac{1 + \frac{3}{5}}{2}} = \sqrt{\frac{\frac{8}{5}}{2}} = \sqrt{\frac{4}{5}} = \frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5} \]

### Step 3: Find \( \tan\left(\frac{x}{2}\right) \)

Using the formula:

\[ \tan\left(\frac{x}{2}\right) = \frac{\sin\left(\frac{x}{2}\right)}{\cos\left(\frac{x}{2}\right)} \]

Substituting the values we found:

\[ \tan\left(\frac{x}{2}\right) = \frac{\frac{\sqrt{5}}{5}}{\frac{2\sqrt{5}}{5}} = \frac{\sqrt{5}}{2\sqrt{5}} = \frac{1}{2} \]

### Final Answers

Thus, the values are:

\[ \sin\left(\frac{x}{2}\right) = \frac{\sqrt{5}}{5}, \quad \cos\left(\frac{x}{2}\right) = \frac{2\sqrt{5}}{5}, \quad \tan\left(\frac{x}{2}\right) = \frac{1}{2} \]