297
4. Establish the identity 2 cot A = cot A2 , tan A2 :
Solution.
[5 marks]
cos A=2 , sin A=2 sin A=2 cos A=2 2A 2A = cossin2A,cossinA 2 2 2 2 cos A cos A = 1 sin A = sin A 2 = 2 cot A = the left side:
The right side =
5. ABC is a triangle, right angled at C . Let a, b, c denote the lengths of the sides opposite angles Ap, B , C respectively. Given that a = 1p, \B = 75 and that tan 75 = 2 + 3, express b and c in the form p + q 3 such p that p = 2 or 2. [8 marks] b Solution.pSince ABC is a right angled triangle, tan B = a . Since a = 1, b = 2 + 3. By the theorem of Pythagoras, c2 = a2 + b2 =p1 + b2 p p = 1 + (2 + 3)2 = 8 + 4 3 = r(2 + 3): p
p
Therefore c = 2 2 + 3p . (As p in problem 2, p we dismiss the psign.) The problem now is to write 2 2 + 3 as p + q 3 where p = 2 or 2. Square both sides p p 4(2 + 3) = p2 + 3q 2 + 2pq 3: If p = 2, then 8 = 4 +p3q 2 (1) p and 4 3 = 4q 3: (2) From (2), q =p 1. This is not consistent with (1). If p = 2, then 8 = 2p+ 3qp2 (3) p and 4 3 = 2 2q 3: (4) p From (4), q = 2 which is consistent with (3) so that p p pp p p p = q = 2 and c = 2 + 2 3 or 2(1 + 3): Check! p c2 = 2(1 + 3)2
p
= 2(4 + 2p 3) = 4(2 + 3):