Question

use the given information to find the exact value of each of the following.
a. \\( \sin 2 \theta \\) b. \\( \cos 2 \theta \\) c. \\( \tan 2 \theta \\)
\\( \sin \theta=-\frac{3}{5}, \theta \\) lies in quadrant iii
a. \\( \sin 2 \theta= \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Find the value of \(\cos\theta\)

Using the identity \(\sin^{2}\theta+\cos^{2}\theta = 1\), we have \(\cos^{2}\theta=1-\sin^{2}\theta\).
Given \(\sin\theta=-\frac{3}{5}\), then \(\cos^{2}\theta=1 - (-\frac{3}{5})^{2}=1-\frac{9}{25}=\frac{16}{25}\).
Since \(\theta\) lies in quadrant III, \(\cos\theta<0\), so \(\cos\theta =-\frac{4}{5}\).

Step2: Calculate \(\sin2\theta\)

Using the double - angle formula \(\sin2\theta = 2\sin\theta\cos\theta\).
Substitute \(\sin\theta=-\frac{3}{5}\) and \(\cos\theta =-\frac{4}{5}\) into the formula:
\(\sin2\theta=2\times(-\frac{3}{5})\times(-\frac{4}{5})\)
\(=\frac{24}{25}\)

Answer:

\(\frac{24}{25}\)