To find [tex]\(\sin \theta\)[/tex] given that [tex]\(\tan \theta = \frac{12}{5}\)[/tex] and [tex]\(\cos \theta = -\frac{5}{13}\)[/tex], follow these steps:
1. Recall the trigonometric identity:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\][/tex]
2. Express [tex]\(\sin \theta\)[/tex] in terms of [tex]\(\tan \theta\)[/tex] and [tex]\(\cos \theta\)[/tex]:
[tex]\[
\sin \theta = \tan \theta \cdot \cos \theta
\][/tex]
3. Substitute the given values:
[tex]\[
\sin \theta = \left( \frac{12}{5} \right) \cdot \left( -\frac{5}{13} \right)
\][/tex]
4. Simplify the expression:
[tex]\[
\sin \theta = \frac{12 \cdot -5}{5 \cdot 13} = \frac{-60}{65}
\][/tex]
5. Simplify the fraction:
[tex]\[
\sin \theta = -\frac{60}{65} = -\frac{12}{13}
\][/tex]
Thus, [tex]\(\sin \theta = -\frac{12}{13}\)[/tex].
Now, let's match this result to the given options:
- A. [tex]\(12\)[/tex]
- B. [tex]\(\frac{5}{13}\)[/tex]
- C. [tex]\(-\frac{5}{13}\)[/tex]
- D. [tex]\(\frac{12}{13}\)[/tex]
- E. [tex]\(-\frac{12}{13}\)[/tex]
The correct answer is indeed [tex]\(\boxed{-\frac{12}{13}}\)[/tex], which corresponds to option E.
