HKDSE MATHEMATICS Ronald Hui Tak Sun Secondary School

HOMEWORK  SHW6-01, 6-A1  Almost all returned. Good job!  SHW6-B1, 6-C1  Deadline: 18 Jan 2016 (Monday)  SHW6-R1, 6-P1  Deadline: 25 Jan 2016 (Monday)  No more delay please 22 October 2015 Ronald HUI

Book 5A Chapter 6 Practical 3-dimensional Problems

In 3-dimensional space, can we use angle of elevation, angle of depression and bearing to describe the relative positions of points?

Yes, we can. N V A BO 45  30  E the angle of elevation of V from A is 30 , the angle of depression of B from V is 45 . For example,

N V A BO 45  30  E N A B O 60  E V 30  U Besides, we can also find the bearings of U and V from O in the figure. the compass bearing of A from O is N30  E, the compass bearing of B from O is S60  E. One more example,

H O horizontal ground In the figure, O is the point on a horizontal ground and H is a point above the ground. If K is the projection of H on the horizontal ground and the bearing of K from O is N  E, then K  N X The bearing of H from O is represented by the bearing of K from O, i.e. N  E. E

In the figure, A, B and C are three points on the horizontal ground. TA is a vertical tower. Then, the true bearing of T from C is ________. N N N 18  65  T A C B the compass bearing of T from B is _________, N65  W 072  90  – 18  = 72 

In the figure, TA is a vertical building of height 30 m. A, B and C are three points on the horizontal ground such that BC = 45 m and ∠ TCA = 40 . C is due west of A and B is due south of A. Find the compass bearing of C from B, correct to 3 significant figures. N T A E B C 40  30 m 45 m Consider △ ACT. tan  TCA AC TA  AC 30 m  tan 40  AC 30  tan 40  m Consider △ ABC. sin  ABC BC AC   ABC 52.6  (cor. to 3 sig. fig.)  45 m  30 tan 40  m ∴ The compass bearing of C from B is N52.6  W.

Follow-up question N 93 m T A 200 m B C 55  70  E In the figure, TC is a vertical building of height 200 m. A, B and C are three points on the horizontal ground. The angle of elevation of T from A is 55 . The compass bearings of A and B from T are N70  E and S80  E respectively. If BC = 93 m, find the distance between A and B, correct to 3 significant figures. Consider △ ACT. tan  CAT AC TC  AC 200 m  tan 55  80   ACB = 180   70   80   adj. ∠ s on st. line = 30  AC 200  tan 55  m

N 93 m T A 200 m B C 55  70  E 80  Consider △ ABC. AB 2 = AC 2 + BC 2 – 2(AC)(BC)cos  ACB ∴ The distance between A and B is 75.5 m. AC 200  tan 55  m  ACB = 30 