Answer
$$a=\sqrt3$$
Work Step by Step
$$\angle A=60^\circ\hspace{.75cm}\angle B=75^\circ\hspace{.75cm}AB=\sqrt2\hspace{.75cm}BC=a$$
1) Analysis:
Here 2 angles and 2 sides are given. Every time 2 angles are given, we can calculate the other angle using the sum of 3 angles in the triangle law. Finally, the unknown side $a$ can be figured out by the law of sines.
2) Calculate the unknown angle $\angle C$
We know that the sum of 3 angles in a triangle is $180^\circ$.
$$\angle A+\angle B+\angle C=180^\circ$$
$$60^\circ+75^\circ+\angle C=180^\circ$$
$$135^\circ+\angle C=180^\circ$$
$$\angle C=180^\circ-135^\circ=45^\circ$$
3) Calculate the unknown side $a$
We know the opposite angle of $a$: $\angle A=60^\circ$. We also know side $AB=\sqrt2$ and its opposite angle $\angle C=45^\circ$.
Therefore, using the law of sines:
$$\frac{a}{\sin A}=\frac{AB}{\sin C}$$
$$a=\frac{AB\sin A}{\sin C}$$
$$a=\frac{\sqrt2\sin60^\circ}{\sin45^\circ}$$
$$a=\frac{\sqrt2\frac{\sqrt3}{2}}{\frac{\sqrt2}{2}}$$
$$a=\frac{\frac{\sqrt6}{2}}{\frac{\sqrt2}{2}}$$
$$a=\frac{\sqrt6\times2}{\sqrt2\times2}=\sqrt3$$
