If `pi < theta < (3pi)/(2)` and `cos theta = -3/5`, then `tan ((theta)/(4))` is equal to

Options

A. `(sqrt(5) - 1)/(2)`
B. `(sqrt(5) + 1)/(2)`
C. `(-sqrt(5) - 1)/(4)`
D. `(-sqrt(5) + 1)/(4)`

Answer

To solve the problem, we need to find the value of \( \tan\left(\frac{\theta}{2}\right) \) given that \( \pi < \theta < \frac{3\pi}{2} \) and \( \cos \theta = -\frac{3}{5} \).

### Step-by-Step Solution:

1. **Identify the Quadrant**:
Since \( \pi < \theta < \frac{3\pi}{2} \), we know that \( \theta \) is in the third quadrant. In this quadrant, both sine and cosine are negative.

**Hint**: Remember the signs of trigonometric functions in different quadrants.

2. **Find \( \sin \theta \)**:
We know that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting \( \cos \theta = -\frac{3}{5} \):
\[ \sin^2 \theta + \left(-\frac{3}{5}\right)^2 = 1 \]
\[ \sin^2 \theta + \frac{9}{25} = 1 \]
\[ \sin^2 \theta = 1 - \frac{9}{25} = \frac{25}{25} - \frac{9}{25} = \frac{16}{25} \]
Taking the square root:
\[ \sin \theta = -\sqrt{\frac{16}{25}} = -\frac{4}{5} \]
(We take the negative root because sine is negative in the third quadrant.)

**Hint**: Use the Pythagorean identity to find sine from cosine.

3. **Calculate \( \tan\left(\frac{\theta}{2}\right) \)**:
We use the half-angle formula for tangent:
\[ \tan\left(\frac{\theta}{2}\right) = \frac{\sin \theta}{1 + \cos \theta} \]
Substituting the values we found:
\[ \tan\left(\frac{\theta}{2}\right) = \frac{-\frac{4}{5}}{1 - \frac{3}{5}} \]
Simplifying the denominator:
\[ 1 - \frac{3}{5} = \frac{5}{5} - \frac{3}{5} = \frac{2}{5} \]
So we have:
\[ \tan\left(\frac{\theta}{2}\right) = \frac{-\frac{4}{5}}{\frac{2}{5}} = -\frac{4}{5} \cdot \frac{5}{2} = -2 \]

4. **Final Result**:
Thus, the value of \( \tan\left(\frac{\theta}{2}\right) \) is:
\[ \tan\left(\frac{\theta}{2}\right) = -2 \]

### Summary:
The final answer is \( \tan\left(\frac{\theta}{2}\right) = -2 \).