If `sintheta=(-4)/(5) and theta` lies in third quadrant, then the value of `cos""(theta)/(2)` is

Options

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

Explanation

To solve the problem, we will follow these steps:

### Step 1: Identify the given information
We are given that \( \sin \theta = -\frac{4}{5} \) and that \( \theta \) lies in the third quadrant.

### Step 2: Determine the range for \( \theta \)
Since \( \theta \) is in the third quadrant, it lies between \( \pi \) and \( \frac{3\pi}{2} \).

### Step 3: Use the Pythagorean identity to find \( \cos \theta \)
We can use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find \( \cos \theta \).

\[ \sin^2 \theta = \left(-\frac{4}{5}\right)^2 = \frac{16}{25} \]

Substituting into the identity:

\[ \frac{16}{25} + \cos^2 \theta = 1 \]

\[ \cos^2 \theta = 1 - \frac{16}{25} = \frac{25}{25} - \frac{16}{25} = \frac{9}{25} \]

Taking the square root:

\[ \cos \theta = \pm \sqrt{\frac{9}{25}} = \pm \frac{3}{5} \]

Since \( \theta \) is in the third quadrant, \( \cos \theta \) is negative:

\[ \cos \theta = -\frac{3}{5} \]

### Step 4: Use the double angle formula to find \( \cos \frac{\theta}{2} \)
We use the formula \( \cos \theta = 2 \cos^2 \frac{\theta}{2} - 1 \).

Substituting \( \cos \theta \):

\[ -\frac{3}{5} = 2 \cos^2 \frac{\theta}{2} - 1 \]

### Step 5: Solve for \( \cos^2 \frac{\theta}{2} \)
Rearranging the equation:

\[ 2 \cos^2 \frac{\theta}{2} = -\frac{3}{5} + 1 \]

\[ 2 \cos^2 \frac{\theta}{2} = -\frac{3}{5} + \frac{5}{5} = \frac{2}{5} \]

Dividing both sides by 2:

\[ \cos^2 \frac{\theta}{2} = \frac{2}{10} = \frac{1}{5} \]

### Step 6: Find \( \cos \frac{\theta}{2} \)
Taking the square root:

\[ \cos \frac{\theta}{2} = \pm \sqrt{\frac{1}{5}} = \pm \frac{1}{\sqrt{5}} \]

### Step 7: Determine the sign of \( \cos \frac{\theta}{2} \)
Since \( \theta \) is in the third quadrant, \( \frac{\theta}{2} \) will be in the second quadrant (as \( \frac{\pi}{2} < \frac{\theta}{2} < \frac{3\pi}{4} \)). In the second quadrant, cosine is negative.

Thus:

\[ \cos \frac{\theta}{2} = -\frac{1}{\sqrt{5}} \]

### Final Answer
The value of \( \cos \frac{\theta}{2} \) is:

\[ -\frac{1}{\sqrt{5}} \]

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