Law of Sines — Examples & Problems

First,

C=180^\circ-30^\circ-45^\circ=105^\circ

a:b:c=\sin30^\circ:\sin45^\circ:\sin105^\circ

We have

\begin{aligned} \sin105^\circ &=\sin(60^\circ+45^\circ)\\ &=\frac{\sqrt6+\sqrt2}{4} \end{aligned}

Therefore,

a:b:c=\frac12:\frac{\sqrt2}{2}:\frac{\sqrt6+\sqrt2}{4}

Multiplying by 4,

a:b:c=2:2\sqrt2:\sqrt6+\sqrt2