Sure, let's solve this problem step by step.
Given:
[tex]\[ \sin \theta = -\frac{7}{25} \][/tex]
and we know that [tex]\( \theta \)[/tex] lies in the third quadrant, that is,
[tex]\[ \pi < \theta < \frac{3\pi}{2} \][/tex]
### Step 1: Determine [tex]\(\cos \theta\)[/tex]
First, recall the Pythagorean identity:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
We are given [tex]\(\sin \theta = -\frac{7}{25}\)[/tex], so:
[tex]\[ \left( -\frac{7}{25} \right)^2 + \cos^2 \theta = 1 \][/tex]
[tex]\[ \left( \frac{49}{625} \right) + \cos^2 \theta = 1 \][/tex]
[tex]\[ \cos^2 \theta = 1 - \frac{49}{625} \][/tex]
[tex]\[ \cos^2 \theta = \frac{625}{625} - \frac{49}{625} \][/tex]
[tex]\[ \cos^2 \theta = \frac{576}{625} \][/tex]
[tex]\[ \cos \theta = \pm \sqrt{\frac{576}{625}} \][/tex]
[tex]\[ \cos \theta = \pm \frac{24}{25} \][/tex]
Since [tex]\(\theta\)[/tex] lies in the third quadrant, both sine and cosine are negative:
[tex]\[ \cos \theta = -\frac{24}{25} \][/tex]
### Step 2: Use the double-angle identity for sine
The double-angle identity for sine is:
[tex]\[ \sin 2\theta = 2 \sin \theta \cos \theta \][/tex]
Substitute the values we have:
[tex]\[ \sin 2\theta = 2 \left( -\frac{7}{25} \right) \left( -\frac{24}{25} \right) \][/tex]
[tex]\[ \sin 2\theta = 2 \cdot \frac{7 \cdot 24}{25 \cdot 25} \][/tex]
[tex]\[ \sin 2\theta = 2 \cdot \frac{168}{625} \][/tex]
[tex]\[ \sin 2\theta = \frac{336}{625} \][/tex]
So, the exact solution for [tex]\(\sin 2 \theta\)[/tex] is:
[tex]\[ \frac{336}{625} \][/tex]
Thus, the correct answer is:
[tex]\[ \frac{336}{625} \][/tex]
