In triangle ABC, side BC = 8 cm, angle A = 30 °, and angle C = 45 °. Find: AB.

Let’s use the theorem of sines: the sides of a triangle are proportional to the sines of the opposite angles. Opposite the side BC lies the angle A, and opposite the side AB – angle C, then:
ВС / sinA = AB / sinC;
8 / sin30 = AB / sin45;
AB = 8 * sin45 / sin30 = 8 * (√2 / 2) / (1/2) = 4√2 * 2 = 8√2 (cm).
Answer: AB = 8√2 cm.