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 = \frac { 24 } { 25 } \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in t
b. \\( \cos 2 \theta = \\)
(simplify you answer, including any radicals. use integers or fractions for any numbers in th

Explanation:

Step1: Find \(\cos\theta\)

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

Step2: Calculate \(\cos2\theta\)

Use the double - angle formula \(\cos2\theta=\cos^{2}\theta-\sin^{2}\theta\).
Substitute \(\sin\theta =-\frac{3}{5}\) and \(\cos\theta=-\frac{4}{5}\) into the formula:
\(\cos2\theta=(-\frac{4}{5})^{2}-(-\frac{3}{5})^{2}=\frac{16}{25}-\frac{9}{25}=\frac{7}{25}\)

Answer:

\(\frac{7}{25}\)