Question

Solve the triangle. Round the side length to the nearest tenth and the angle measures to the nearest degree.

Answer 1

Answer: 41, 54, 6.1

Explanation:

By the law of Cosines,

`b^2=4^2 +5^2 -2(4)(5)\cos 85^(\circ)\\\\b=\sqrt{4^2 +5^2 -2(4)(5)\cos 85^(\circ)} \\\\ =\sqrt{41-40 \cos 85^(\circ)}\\\\\approx 6.1`

By the law of sines,

`(\sin A)/(a)=(\sin C)/(c)=(\sin B)/(b)\\\\(\sin A)/(4)=(\sin C)/(5)=\frac{\sin 85^(\circ)}{\sqrt{41-40 \cos (85^(\circ))}}\\\\\implies \sin A=\frac{4\sin 85^(\circ)}{\sqrt{41-40 \cos 85^(\circ)}} \implies \angle A \approx 41^(\circ)\\\\\implies \sin C=\frac{5\sin 85^(\circ)}{\sqrt{41-40 \cos 85^(\circ)}} \implies \angle C \approx 54^(\circ)`