If `sin theta = - (3)/(5) and theta` lies in the third quadrant, then the value of cos `(theta//2)`is

Options

A. `(1)/(5)`
B. `(-1)/(sqrt ((10))`
C. `(-1)/(sqrt5)`
D. `(1)/( sqrt ((10))`

Answer

To find the value of \( \cos \left( \frac{\theta}{2} \right) \) given that \( \sin \theta = -\frac{3}{5} \) and \( \theta \) lies in the third quadrant, we can follow these steps:

### Step 1: Use the Pythagorean Identity
We start with the identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting the value of \( \sin \theta \):
\[ \left(-\frac{3}{5}\right)^2 + \cos^2 \theta = 1 \]
Calculating \( \left(-\frac{3}{5}\right)^2 \):
\[ \frac{9}{25} + \cos^2 \theta = 1 \]

### Step 2: Solve for \( \cos^2 \theta \)
Rearranging the equation:
\[ \cos^2 \theta = 1 - \frac{9}{25} \]
Finding a common denominator:
\[ \cos^2 \theta = \frac{25}{25} - \frac{9}{25} = \frac{16}{25} \]

### Step 3: Find \( \cos \theta \)
Since \( \theta \) is in the third quadrant, where cosine is negative:
\[ \cos \theta = -\sqrt{\frac{16}{25}} = -\frac{4}{5} \]

### Step 4: Use the Half-Angle Formula
We use the half-angle formula for cosine:
\[ \cos \left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{1 + \cos \theta}{2}} \]
Substituting the value of \( \cos \theta \):
\[ \cos \left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{1 - \frac{4}{5}}{2}} \]
Calculating the expression inside the square root:
\[ \cos \left( \frac{\theta}{2} \right) = \pm \sqrt{\frac{\frac{1}{5}}{2}} = \pm \sqrt{\frac{1}{10}} = \pm \frac{1}{\sqrt{10}} \]

### Step 5: Determine the Sign of \( \cos \left( \frac{\theta}{2} \right) \)
Since \( \theta \) is in the third quadrant, \( \frac{\theta}{2} \) will be in the second quadrant, where cosine is negative:
\[ \cos \left( \frac{\theta}{2} \right) = -\frac{1}{\sqrt{10}} \]

### Final Answer
Thus, the value of \( \cos \left( \frac{\theta}{2} \right) \) is:
\[ \cos \left( \frac{\theta}{2} \right) = -\frac{1}{\sqrt{10}} \]