If `sinalpha=-(3)/(5)` and `alpha` lies in the third quadrant then find the value of `cos.(alpha)/(2)`

Answer

To solve the problem, we need to find the value of \(\cos\left(\frac{\alpha}{2}\right)\) given that \(\sin\alpha = -\frac{3}{5}\) and that \(\alpha\) lies in the third quadrant.

### Step 1: Understand the given information
We know that:
- \(\sin\alpha = -\frac{3}{5}\)
- \(\alpha\) is in the third quadrant.

### Step 2: Draw a right triangle
In the third quadrant, both sine and cosine are negative. We can represent \(\sin\alpha\) as:
\[ \sin\alpha = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{-3}{5} \]
This means that the opposite side (height) is -3 (negative because it's below the x-axis) and the hypotenuse is 5.

### Step 3: Use the Pythagorean theorem to find the adjacent side
Using the Pythagorean theorem:
\[ \text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2 \]
Substituting the known values:
\[ 5^2 = (-3)^2 + \text{adjacent}^2 \]
\[ 25 = 9 + \text{adjacent}^2 \]
\[ \text{adjacent}^2 = 25 - 9 = 16 \]
\[ \text{adjacent} = -4 \quad (\text{negative because it's in the third quadrant}) \]

### Step 4: Calculate \(\cos\alpha\)
Now, we can find \(\cos\alpha\):
\[ \cos\alpha = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{-4}{5} \]

### Step 5: Use the half-angle formula for cosine
We know the half-angle formula:
\[ \cos\left(\frac{\alpha}{2}\right) = \sqrt{\frac{1 + \cos\alpha}{2}} \]
Substituting \(\cos\alpha = -\frac{4}{5}\):
\[ \cos\left(\frac{\alpha}{2}\right) = \sqrt{\frac{1 - \frac{4}{5}}{2}} = \sqrt{\frac{\frac{1}{5}}{2}} = \sqrt{\frac{1}{10}} = \frac{1}{\sqrt{10}} \]

### Step 6: Determine the sign of \(\cos\left(\frac{\alpha}{2}\right)\)
Since \(\alpha\) is in the third quadrant, \(\frac{\alpha}{2}\) will be in the second quadrant (as it ranges from \(\frac{\pi}{2}\) to \(\frac{3\pi}{4}\)). In the second quadrant, cosine is negative:
\[ \cos\left(\frac{\alpha}{2}\right) = -\frac{1}{\sqrt{10}} \]

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