Trigonometry (11th Edition) Clone

$$a=\sqrt3$$
$$\angle A=60^\circ\hspace{.75cm}\angle B=75^\circ\hspace{.75cm}AB=\sqrt2\hspace{.75cm}BC=a$$ 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: $$\frac{a}{\sin A}=\frac{AB}{\sin C}$$ $$a=\frac{AB\sin A}{\sin C}$$ $$a=\frac{\sqrt2\sin60^\circ}{\sin45^\circ}$$ $$a=\frac{\sqrt2\frac{\sqrt3}{2}}{\frac{\sqrt2}{2}}$$ $$a=\frac{\frac{\sqrt6}{2}}{\frac{\sqrt2}{2}}$$ $$a=\frac{\sqrt6\times2}{\sqrt2\times2}=\sqrt3$$