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 the e
b. \\( \cos 2 \theta= \\) \\( \frac{7}{25} \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the exp
c. \\( \tan 2 \theta= \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expr

Explanation:

Step1: Recall the double - angle formula for tangent

The formula for \(\tan2\theta=\frac{\sin2\theta}{\cos2\theta}\).

Step2: Substitute the values of \(\sin2\theta\) and \(\cos2\theta\)

We know that \(\sin2\theta = \frac{24}{25}\) and \(\cos2\theta=\frac{7}{25}\).
So, \(\tan2\theta=\frac{\frac{24}{25}}{\frac{7}{25}}\).

Step3: Simplify the expression

When dividing by a fraction, we multiply by its reciprocal. So \(\tan2\theta=\frac{24}{25}\times\frac{25}{7}\).
The \(25\) terms cancel out, and we get \(\tan2\theta=\frac{24}{7}\).

Answer:

\(\frac{24}{7}\)