use identities to find the values of the sine and cosine functions for the following angle measure.\nθ…

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{12}{13}), we substitute it into the identity: (\frac{12}{13}=1 - 2\sin^{2}\theta).

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

First, rearrange the equation (\frac{12}{13}=1 - 2\sin^{2}\theta) to get (2\sin^{2}\theta=1-\frac{12}{13}). Calculate (1-\frac{12}{13}=\frac{13 - 12}{13}=\frac{1}{13}). So, (2\sin^{2}\theta=\frac{1}{13}), then (\sin^{2}\theta=\frac{1}{26}).

Step3: Solve for (\sin\theta)

Since (\theta) terminates in quadrant I, (\sin\theta>0). Take the square root of (\sin^{2}\theta=\frac{1}{26}), (\sin\theta=\sqrt{\frac{1}{26}}=\frac{\sqrt{1}}{\sqrt{26}}=\frac{1}{\sqrt{26}}). Rationalize the denominator: (\sin\theta=\frac{1\times\sqrt{26}}{\sqrt{26}\times\sqrt{26}}=\frac{\sqrt{26}}{26}).

Answer:

(\frac{\sqrt{26}}{26})