Double and Half angle formulas

1. Find $\sin (2 x), \cos (2 x)$, and $\tan (2 x)$ from the given information.   solution

  $\sin (x)=\frac{5}{13}, \quad x \text { in Quadrant I}$

2. If $\cos (x)=-\frac{4}{5}$ and $\sin(x)>0$. Find the exact values of $\sin (2 x), \cos (2 x)$, and $\tan (2 x)$.   solution

3. If $\cos (x)= \frac{5}{13}$ and $x$ is in Quadrant I, find the value of $\sin \frac{x}{2}$. solution

4. If \( \cos \alpha = \frac{5}{13}, \frac{3\pi}{2}< \alpha < 2\pi \). Find the exact value of \( \tan (2\alpha)\).   solution

5. If \(\sin \theta = -\frac{\sqrt 3}{3},  \frac{3\pi}{2}< \theta < 2\pi \). Find the exact value of \( \cos \frac{\theta}{2}\).   solution