Answer
$$\sin B=\frac{3\sqrt{21}}{14}$$
Work Step by Step
$$a = 2 \hspace{1cm}b=3\hspace{1cm}C=60^\circ$$
- From Exercise 61, the law of sines: $$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$$
To find $\sin B$, we still need to find $c$. And we can use the law of cosines to do that:
$$c^2=a^2+b^2-2ab\cos C$$
Therefore,
$$c^2=2^2+3^2-2\times2\times3\times\cos60^\circ$$
$$c^2=4+9-12\times\frac{1}{2}$$
$$c^2=13-6$$
$$c^2=7$$
$$c=\sqrt7$$ (as $c\gt0$)
- Now to find $\sin B$, we use the law of sines:
$$\frac{\sin B}{b}=\frac{\sin C}{c}$$
$$\sin B=\frac{b\times\sin C}{c}$$
$$\sin B=\frac{3\times\sin60^\circ}{\sqrt7}$$
$$\sin B=\frac{3\times\frac{\sqrt3}{2}}{\sqrt7}$$
$$\sin B=\frac{3\sqrt3}{2\sqrt7}$$
$$\sin B=\frac{3\sqrt{21}}{14}$$
