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.

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 ：\(\boxed{A \approx 30^\circ, B \approx 61^\circ, C \approx 89^\circ, a = 24, b = 42, c \approx 48}\)