(2) Given $\sin (\alpha)=\frac{2}{\sqrt{7}}$ and $\sin (\beta)=\frac{\sqrt{2}}{3}$, evaluate the following:

A$\sin (\alpha-\beta)$

B$\cos (\alpha+\beta)$

C$\tan (\alpha-\beta)$

D$\sin (2 \alpha)$

E$\cos (2 \alpha)$

(f) $\cos (2 \beta)$



(3) Given $\sin (\alpha)=-\frac{\sqrt{6}}{5}$ and $\cos (\beta)=-\frac{4}{\sqrt{17}}$, with $\alpha$ in QIV and $\beta$ in QII, evaluate the following:

A$\sin (\alpha+2 \beta)$

B$\cos (2 \alpha-\beta)$

C$\cos (2 \alpha-2 \beta)$

(Hint: Using the sum identities, $\sin (\alpha+2 \beta)=\sin \alpha \cos (2 \beta)+ \cos \alpha \sin (2 \beta)$, to compute $\sin (\alpha+2 \beta)$, what values do you need to evaluate first?)



(4) To use the half-angle identities effectively, we need to know which quadrant the angle $\frac{\alpha}{2}$ belongs to, provided the quadrant of the angle $\alpha$.