The Laws of Sines and Cosines · Paper 3 — Free Maths AI HL Practice Questions

A second surveyor, working from a different reference, models a triangle $PQR$ in which $PQ = 120\,\text{m}$, $QR = 150\,\text{m}$, angle $Q = 36.9°$, and the side opposite angle $P$ is $PR = d\,\text{m}$. Using the Law of Sines applied to triangle $ABC$ from part (a), with the known angle $B$ and its opposite side $CA = 90\,\text{m}$, find the two possible values of angle $A$, correct to one decimal place. [4 marks]

Justify why the Law of Sines produces two candidate values for angle $A$, and evaluate which value is geometrically valid for triangle $ABC$. Support your reasoning with reference to the angle sum of a triangle and the relative lengths of the sides. Clarification of labelling for (b): In triangle $ABC$, angle $B \approx 36.9°$ is opposite side $CA = 90\,\text{m}$. Side $BC = 150\,\text{m}$ is opposite angle $A$. Apply the Law of Sines to find angle $A$. [3 marks]