2.33: Exact tan and hypotenuse with surds

0606/23/O/N/19 — Question 6 · 6 marks

Do not use a calculator in this question.

2.33 diagram
In the right-angled triangle ABCABC, ∠B=90°\angle B = 90^{\degree}, AB=3+3AB = 3 + \sqrt{3} and BC=3−3BC = 3 - \sqrt{3}.
(i)
Find tan⁡∠ACB\tan \angle ACB in the form r+s3r + s\sqrt{3}, where rr and ss are integers.
[3]
(ii)
Find ACAC in the form tut\sqrt{u}, where tt and uu are integers and t≠1t \neq 1.
[3]