Question

Three deer, A, B, and C, are grazing in a field. Deer B is located 62m from deer A at an angle of 51° north of West. Deer C is located 77° north of east relative to deer A. The distance between deer B and C is 95m. What is the distance between deer A and C?

Answer 1

Set deer A's position to be the origin. Let $c$ be the distance from deer A to deer C. We're given that deer B is 95 m away from deer C, which means the length of the vector $B-C$ is 95 (or $C-B$). Then

$|B-C|^2=(B-C)\cdot(B-C)=B\cdot B-2B\cdot C+C\cdot C=|B|^2-2B\cdot C+|C|^2$

$|B-C|^2=|B|^2-2|B||C|\cos(180-77-51)^\circ+c^2$

$95^2=62^2-2(62)(c)\cos52^\circ+c^2$

$c^2-124\cos52^\circ c-5181=0\implies c=120\,\mathrm m$