Use the Sine Law in an Ambiguous Case
In $\triangle \mathrm{ABC}, \angle \mathrm{A}=30^{\circ}, a=24 \mathrm{~cm}$, and $b=42 \mathrm{~cm}$. Determine the measures of the other side and angles. Round your answers to the nearest unit.
\(\boxed{A \approx 30^\circ, B \approx 61^\circ, C \approx 89^\circ, a = 24, b = 42, c \approx 48}\)
Step 1 :\(\sin A = \frac{1}{2}\)
Step 2 :\(\frac{a}{\sin A} = \frac{b}{\sin B}\)
Step 3 :\(\frac{24}{\frac{1}{2}} = \frac{42}{\sin B}\)
Step 4 :\(\sin B = \frac{42}{48} = \frac{7}{8}\)
Step 5 :\(B = \sin^{-1} \frac{7}{8}\)
Step 6 :\(B \approx 61^\circ\)
Step 7 :\(C = 180^\circ - A - B \approx 180^\circ - 30^\circ - 61^\circ \approx 89^\circ\)
Step 8 :\(\frac{c}{\sin C} = \frac{a}{\sin A}\)
Step 9 :\(\frac{c}{\sin 89^\circ} = \frac{24}{\frac{1}{2}}\)
Step 10 :\(c \approx 48\)
Step 11 :\(\boxed{A \approx 30^\circ, B \approx 61^\circ, C \approx 89^\circ, a = 24, b = 42, c \approx 48}\)