use identities to find values of the sine and cosine functions for the angle measure. \nθ, given that ( cos…

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

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

Answer

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})