Question

use identities to find \\(\sin \theta\\) if \\(\cos 2\theta = -\frac{4}{7}\\) and \\(\frac{\pi}{2} < \theta < \pi\\).

\\(\bigcirc\\) a. \\(\frac{\sqrt{154}}{14}\\)

\\(\bigcirc\\) b. \\(-\frac{\sqrt{154}}{14}\\)

\\(\bigcirc\\) c. \\(\frac{1}{14}\\)

\\(\bigcirc\\) d. \\(-\frac{1}{14}\\)

Explanation:

Step1: Use the double - angle identity

The double - angle identity for cosine is \(\cos2\theta = 1 - 2\sin^{2}\theta\).
Given \(\cos2\theta=-\frac{4}{7}\), we substitute it into the identity:
\(-\frac{4}{7}=1 - 2\sin^{2}\theta\).

Step2: Solve for \(\sin^{2}\theta\)

First, rearrange the equation \(-\frac{4}{7}=1 - 2\sin^{2}\theta\) to isolate \(\sin^{2}\theta\).
Subtract \(1\) from both sides: \(-\frac{4}{7}-1=- 2\sin^{2}\theta\).
\(-\frac{4 + 7}{7}=-2\sin^{2}\theta\), so \(-\frac{11}{7}=-2\sin^{2}\theta\).
Then \(\sin^{2}\theta=\frac{11}{14}\).

Step3: Determine the sign of \(\sin\theta\)

Since \(\frac{\pi}{2}<\theta<\pi\), \(\theta\) is in the second quadrant. In the second quadrant, \(\sin\theta>0\).
\(\sin\theta=\sqrt{\frac{11}{14}}=\frac{\sqrt{11\times14}}{14}=\frac{\sqrt{154}}{14}\).

Answer:

A. \(\frac{\sqrt{154}}{14}\)