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 need to find the value of \(\cos\left(\frac{\theta}{2}\right)\) given that \(\sin\theta = -\frac{4}{5}\) and \(\theta\) lies in the third quadrant. Here are the steps to find the solution:

### Step 1: Identify \(\cos\theta\)
Since \(\sin\theta = -\frac{4}{5}\) and \(\theta\) is in the third quadrant, we can use the Pythagorean identity:
\[ \sin^2\theta + \cos^2\theta = 1 \]
Substituting the value of \(\sin\theta\):
\[ \left(-\frac{4}{5}\right)^2 + \cos^2\theta = 1 \]
\[ \frac{16}{25} + \cos^2\theta = 1 \]
\[ \cos^2\theta = 1 - \frac{16}{25} \]
\[ \cos^2\theta = \frac{25}{25} - \frac{16}{25} = \frac{9}{25} \]
Taking the square root, since \(\theta\) is in the third quadrant where cosine is negative:
\[ \cos\theta = -\sqrt{\frac{9}{25}} = -\frac{3}{5} \]

### Step 2: Use the Half-Angle Formula
Now we will use the half-angle formula for cosine:
\[ \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos\theta}{2}} \]
Substituting \(\cos\theta = -\frac{3}{5}\):
\[ \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \frac{3}{5}}{2}} \]
\[ = \sqrt{\frac{\frac{5}{5} - \frac{3}{5}}{2}} = \sqrt{\frac{\frac{2}{5}}{2}} = \sqrt{\frac{2}{10}} = \sqrt{\frac{1}{5}} = \frac{1}{\sqrt{5}} \]

### Step 3: 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 (as it is half of an angle in the third quadrant). In the second quadrant, cosine is negative:
\[ \cos\left(\frac{\theta}{2}\right) = -\frac{1}{\sqrt{5}} \]

### Final Answer
Thus, the value of \(\cos\left(\frac{\theta}{2}\right)\) is:
\[ \cos\left(\frac{\theta}{2}\right) = -\frac{1}{\sqrt{5}} \]

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