Use identities to find values of the sine and cosine functions for the angle measure.
$\theta$, given that $\cos 2 \theta=\frac{12}{13}$ and $90^{\circ}< \theta< 180^{\circ}$
Answer
Final Answer: The values of the sine and cosine functions for the angle measure \(\theta\) are \(\sin \theta = \boxed{-0.196}\) and \(\cos \theta = \boxed{0.981}\).
Step 1 ：We are given the value of \(\cos 2 \theta = \frac{12}{13}\) and we need to find the values of \(\sin \theta\) and \(\cos \theta\).
Step 2 ：We can use the double angle identities for cosine and sine to solve this problem. The double angle identity for cosine is \(\cos 2 \theta = 1 - 2 \sin^2 \theta = 2 \cos^2 \theta - 1\).
Step 3 ：Since \(\cos 2 \theta\) is positive and \(90^\circ<\theta<180^\circ\), \(\sin \theta\) must be negative.
Step 4 ：We can solve for \(\sin \theta\) using the identity \(\cos 2 \theta = 1 - 2 \sin^2 \theta\) and then find \(\cos \theta\) using the Pythagorean identity \(\cos^2 \theta = 1 - \sin^2 \theta\).
Step 5 ：By solving, we get \(\sin \theta = -0.19611613513818396\) and \(\cos \theta = 0.9805806756909202\).
Step 6 ：Rounding to three decimal places, we get \(\sin \theta = -0.196\) and \(\cos \theta = 0.981\).
Step 7 ：Final Answer: The values of the sine and cosine functions for the angle measure \(\theta\) are \(\sin \theta = \boxed{-0.196}\) and \(\cos \theta = \boxed{0.981}\).
