1.14: Cosine rule with surd lengths

0606/12/M/J/18 — Question 10 · 4 marks

Do not use a calculator in this question. All lengths are in centimetres.

The diagram shows triangle $ABC$, where $AB = 4\sqrt{3} - 5$, $BC = 4\sqrt{3} + 5$ and $\angle ABC = 60^{\degree}$.

It is known that $\sin 60^{\degree} = \tfrac{\sqrt{3}}{2}$, $\cos 60^{\degree} = \tfrac{1}{2}$, $\tan 60^{\degree} = \sqrt{3}$.