To solve for [tex]\(\cos \theta\)[/tex] given that [tex]\(\sin \theta = -\frac{3}{5}\)[/tex] in the third quadrant, we can follow these steps:
1. Identify the given sine value and quadrant:
We know that:
[tex]\[
\sin \theta = -\frac{3}{5}
\][/tex]
The angle [tex]\(\theta\)[/tex] is in the third quadrant. In the third quadrant, both sine and cosine values are negative.
2. Use the Pythagorean identity:
The Pythagorean identity states that:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
3. Calculate [tex]\(\sin^2 \theta\)[/tex]:
Given [tex]\(\sin \theta = -\frac{3}{5}\)[/tex]:
[tex]\[
\sin^2 \theta = \left(-\frac{3}{5}\right)^2 = \frac{9}{25}
\][/tex]
4. Substitute [tex]\(\sin^2 \theta\)[/tex] into the Pythagorean identity to solve for [tex]\(\cos^2 \theta\)[/tex]:
[tex]\[
\frac{9}{25} + \cos^2 \theta = 1
\][/tex]
5. Solve for [tex]\(\cos^2 \theta\)[/tex]:
Subtract [tex]\(\frac{9}{25}\)[/tex] from both sides:
[tex]\[
\cos^2 \theta = 1 - \frac{9}{25} = \frac{25}{25} - \frac{9}{25} = \frac{16}{25}
\][/tex]
6. Determine [tex]\(\cos \theta\)[/tex]:
Taking the square root of [tex]\(\cos^2 \theta\)[/tex]:
[tex]\[
\cos \theta = \pm \sqrt{\frac{16}{25}} = \pm \frac{4}{5}
\][/tex]
Since [tex]\(\theta\)[/tex] is in the third quadrant, the cosine value must be negative:
[tex]\[
\cos \theta = -\frac{4}{5}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{-\frac{4}{5}}
\][/tex]
Hence option (A) is the right one.
