Question

use identities to find values of the sine and cosine functions for the angle measure.
θ, given that ( cos 2\theta=\frac{56}{65} ) and ( 90^{circ}<\theta<180^{circ} )
( sin \theta=square )
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)

Explanation:

Step1: Use the double - angle identity for cosine

The double - angle identity for cosine is \(\cos2\theta = 1 - 2\sin^{2}\theta\).
Given \(\cos2\theta=\frac{56}{65}\), we substitute it into the identity:
\(\frac{56}{65}=1 - 2\sin^{2}\theta\)

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

First, rearrange the equation:
\(2\sin^{2}\theta=1-\frac{56}{65}\)
\(2\sin^{2}\theta=\frac{65 - 56}{65}=\frac{9}{65}\)
Then \(\sin^{2}\theta=\frac{9}{130}\)

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

Since \(90^{\circ}<\theta<180^{\circ}\), \(\theta\) is in the second quadrant. In the second quadrant, \(\sin\theta> 0\)
So \(\sin\theta=\sqrt{\frac{9}{130}}=\frac{3}{\sqrt{130}}=\frac{3\sqrt{130}}{130}\)

Answer:

\(\frac{3\sqrt{130}}{130}\)