Question3. Given sec $x=\frac{\sqrt{10}}{3}$, where $\frac{3 \pi}{2} \leq x \leq 2 \pi$, determine the value of $\cos (2 x)$ (u) $\frac{1}{19}$ B) $-\frac{37}{19}$ C) $\frac{37}{19}$
4. What is the exact value of $\sin \frac{13 \pi}{12}$ ? D) $-\frac{1}{19}$ A) $\frac{\sqrt{6}+\sqrt{2}}{4}$ B) $\frac{\sqrt{6}-\sqrt{2}}{4}$ C) $\frac{\sqrt{2}-\sqrt{6}}{4}$ D) $\frac{\sqrt{2}-\sqrt{3}}{2}$

Studdy Solution
Find $\sin \frac{13\pi}{12}$ using angle subtraction identities:
$\frac{13\pi}{12} = \pi + \frac{\pi}{12}$
Use the identity $\sin(\pi + \theta) = -\sin(\theta)$:
$\sin \frac{13\pi}{12} = -\sin \frac{\pi}{12}$
Using the identity $\sin \frac{\pi}{12} = \sin(15^\circ) = \frac{\sqrt{6} - \sqrt{2}}{4}$:
$\sin \frac{13\pi}{12} = -\frac{\sqrt{6} - \sqrt{2}}{4}$
The value of $\cos(2x)$ is:
$$\boxed{\frac{4}{5}}$$
The exact value of $\sin \frac{13\pi}{12}$ is:
$$\boxed{-\frac{\sqrt{6} - \sqrt{2}}{4}}$$