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Name of Lecturer: Mr. J. Agius Course: HVAC 2 Lesson 35 Chapter 4. Trigonometry Trigonometry in 3D To find angles or lengths in solid figures, a right-angled triangle containing the unknown quantity has to be identified. This triangle should then be drawn separately from the solid figure. E Example 1 The figure is a pyramid on a square base ABCD. The edges of the base are 30cm long and the height EH of the pyramid is 42cm. Find C a) the length of AC b) the angle EAH. D Answer H B A a) Therefore x2 = 900 + 900 = 1800 x = 42.42 Therefore AC = 42.4 cm correct to 3 s.f. b) C AC is diagonal of the square base ABCD. Using ABC, B = 90o, AB = BC = 30cm AC2 = AB2 + BC2 (Pythagoras theorem) x A B 30 EAH is in AHE, in which H = 90o and AH = ½ AC = 21.21 cm E 42 (opp) opp 42 tan A 1.9802 adj 21.21 A = 63.2o 30 A 21.21 H (adj) 4 Trigonometry Page 1 Name of Lecturer: Mr. J. Agius Course: HVAC 2 The Angle Between a Line and a Plane The angle between the line PQ and the plane ABCD is defined as the angle between PQ and its projection on the plane ABCD. Q Q B B C C A A P P N D D Draw a perpendicular, QN, from Q to the plane. (N is called the foot of the perpendicular.) Join P to N. The required angle is QPN. (This angle is tucked under the line.) The line PN is called the projection of the line PQ on the plane. (Note: if the plane does not look horizontal, it may help you to see which line is the perpendicular if you turn the page and look at the diagram from a different angle.) The Angle Between Two Planes P Q To find the angle between two planes we need to find two lines to act as the arms of the angle. The two lines, one in each plane, must meet on the joining line, PQ, of the two planes and each must be at right angles to the joining line. A C P Q One possible angle is ARC. Notice that there are any number of pairs of lines that meet at a given point on the joining line but only a pair at right angles to PQ gives the required angle. DEF is not a possible angle but DQF is possible. 4 Trigonometry Page 2 Name of Lecturer: Mr. J. Agius Course: HVAC 2 D F P E Q It can be helpful to use a section through a solid when trying to identify the angle between two of its faces. The section, or cut, must be made at right angles to the edge where the two faces meet, cutting this edge at P, say. Then the angle between the faces is the angle at Q in the section. For example, in a right pyramid, the angle between the base and a sloping face can be found from a section formed by a cut through the vertex perpendicular to the base. Example 2 ABCD is a right pyramid of height 5cm. Its base ABCD is a rectangle in which AB = 6cm and BC = 8cm. a) Find the angle between EA and the plane ABCD. E b) Find the angle between the planes EBC and ABCD. Answer a) (EF is the perpendicular from E to ABCD) EAF is the required angle. 1 AF AC A 2 In ABC, AC = 10cm (3, 4, 5) Therefore AF = 5cm Therefore EAF = 45o (rt-angled isos. ) D F G 6cm C 8cm B The angle between EA and ABCD is 45o. b) (BC is the joining line for the planes. Form symmetry, EG and FG are both perpendicular to BC at G.) EGF is the required angle. In EFG, F = 90o. 5cm E opp 5 1.6666... adj 3 5cm (opp) EGF = tan 1 (1.6666...) 59 o tan G So E A F The angle between the planes is 59o. F 4 Trigonometry G Page 3 Name of Lecturer: Mr. J. Agius Course: HVAC 2 Exercise 1 V 1) For this pyramid, whose height is 10m, a) Calculate the length of DB b) Calculate the length of VB c) Find the angle that 10m X 10m VB makes with the base ABCD d) C 10m D Calculate the length of VY, where B A Y Y is the midpoint of AB e) Draw the section containing the angle between VY and the face ABCD. f) Calculate the angle described in (e) g) Find the angle between the planes VAB and ABCD. E 2) 13 cm F D C For this wedge, A 20 cm 12 cm a) calculate the lengths of AF and FC b) draw the section containing the angle between the diagonal FC and the ABCD c) find the size of the angle that FC makes with the base ABCD. d) find the size of the angle between the planes FBCE and ABCD. 3) In this cuboid, AB = 7cm, AE = 6cm B base H G and BC = 10cm. Find the size of the angle between a) EC and the plane ABCD b) FD and the plane ABFE c) the planes EBCH and AEHD d) the planes EFCD and CDHG. 4 Trigonometry E F 6 cm D C 10 cm A 7 cm B Page 4 Name of Lecturer: Mr. J. Agius Course: HVAC 2 4) E D Q A N P C B The great pyramid of Cephren at Gizeh in Egypt has a square base of side 215m and is 225m high. In the diagram E is the vertex of the right pyramid and ABCD is its base. The diagonals of the base intersect at N; P and Q are the midpoints of AB and DC respectively. a) A tunnel runs from the entrance P to the burial chamber at N. Find PN. How far is N from A? b) How far is it i) from P to the top ii) from A to the top? c) A climber wishes to climb from the base of the pyramid to the top. Where should he start i) to make the shortest climb ii) to climb in one straight line at the smallest angle to the horizontal? 5) The diagram shows a part of a sea wall whose constant cross-section is a right-angled triangle. The road is at the level of the top, DC, and the beach starts at AB. Jim clambers from the road to the beach straight down the pat DA whereas Pete goes along the path DB. Find a) C D 3m B the length of each path. 8m 4m b) the inclination to the horizontal of each path. A A 6) The diagram shows a corner block of a concrete edging system. AB = BC = BD = 6cm. Find the angle that the sloping face, ACD, makes with the horizontal base, BCD. D B C 4 Trigonometry Page 5