MATHEMATICS GRADE 1 SESSION 18 (LEARNER NOTES) TOPIC 1: TWO-DIMENSIONAL TRIGONOMETRY Learner Note: Before attempting to do any complex two or three dimensional problems involving trigonometry, it is essential for you to do basic questions first using the area, sine and cosine rules. Once you know when to use the rules at a basic level, you will then be able to handle the more complicated problems more effectively. SECTION A: TYPICAL EXAM QUESTIONS QUESTION 1: 0 minutes A piece of land has the form of a quadrilateral ABCD with AB 0m, BC 1m, CD 7m and AD 8m. ˆB 110. The owner decides to divide the land into two plots by erecting a fence from A to C. 110 (a) Calculate the length of the fence AC correct to one decimal place. () (b) Calculate the size of BAC ˆ correct to the nearest degree. () (c) Calculate the size of ˆD, correct to the nearest degree. (3) (d) Calculate the area of the entire piece of land ABCD, correct to one decimal place. (3) [10] QUESTION ABC is an isosceles triangle with AB BCm, AB c, AC b and BC a. b Prove that cosb 1 [4] a Page 1 of 6
MATHEMATICS GRADE 1 SESSION 18 (LEARNER NOTES) TOPIC : THREE-DIMENSIONAL TRIGONOMETRY QUESTION 1: 5 minutes In the diagram below, AB is a straight line 1 500 m long. DC is a vertical tower 158 metres high with C, A and B points in the same horizontal plane. The angles of elevation of D from A and B are 5 and. CAB ˆ 30. 5 30 (a) Determine the length of AC. (3) (b) Find the value of. (5) (c) Calculate the area of ABC. ) (d) Calculate the size of ADB ˆ (6) [16] SECTION B: ADDITIONAL CONTENT NOTES Solving two-dimensional problems using the sine, cosine and area rules The sine-rule can be used when the following is known in the triangle: - more than 1 angle and a side - sides and an angle (not included) sin A sin B sin C a b c The cosine-rule can be used when the following is known of the triangle: - 3 sides - sides and an included angle a b c bc cosa The area of any triangle can be found when at least two sides an included angle are known 1 Area of ABC sin C ab Page of 6
MATHEMATICS GRADE 1 SESSION 18 (LEARNER NOTES) SECTION C: HOMEWORK TOPIC 1: TWO-DIMENSIONAL TRIGONOMETRY QUESTION 1 Two ships, A and B, are 10 km apart. Ship A is at a bearing of 67 from D and 97 km away from D. DN points due north. Ship B is at a bearing of 08 from D. 67 08 (a) Determine the bearing of Ship A from Ship B, that is, ˆ MBA, when BM DN. (6) (b) If Ship B travels due north, and Ship B travels due south, then at some instant of time, Ship A is due east of Ship B. Calculate the distance between the two ships at that instant. (3) [9] QUESTION If b c and a 7b, show why it is impossible to construct ABC. [4] Page 3 of 6
MATHEMATICS GRADE 1 SESSION 18 (LEARNER NOTES) TOPIC : THREE-DIMENSIONAL TRIGONOMETRY QUESTION 1 Thandi is standing at point P on the horizontal ground and observes two poles, AC and BD, of different heights. P, C and D are in the same horizontal plane. From P the angles of inclination to the top of the poles A and B are 3 and 18 respectively. Thandi is 18 m from the base of pole AC. The height of pole BD is 7 m. B A 7 m D 18 C 18 m 3 4 P Calculate, correct to TWO decimal places: (a) The distance from Thandi to the top of pole BD. () (b) The distance from Thandi to the top of pole AC. () (c) The distance between the tops of the poles, that is the length of AB, if APB ˆ 4 (3) [7] QUESTION A rectangular block of wood has a breadth of 6 metres, height of 8 metres and a length of 15 metres. A plane cut is made through the block as shown in the diagram revealing the triangular plane that has been formed. Calculate the size of EBG ˆ. [5] H G E D C 8 cm A 6 cm B 15 cm Page 4 of 6
MATHEMATICS GRADE 1 SESSION 18 (LEARNER NOTES) SECTION D: SOLUTIONS AND HINTS TO SECTION A TOPIC 1: TWO-DIMENSIONAL TRIGONOMETRY 1(a) AC (1 m) (0 m) (1 m)(0 m) cos110 1(b) AC 708,1696688 AC 6, 6m sin BAC ˆ sin110 1m 6, 6m ˆ 1sin110 sin BAC 6,6m sin BAC ˆ 0, 43914831 BAC ˆ 5 OR (1 m) (0 m) (6, 6 m) (0 m)(6, 6 m) cos BAC ˆ ˆ 1064 cos BAC 963,56m cos BAC ˆ 0,9056015038 BAC ˆ 5 1(c) (6,6 m) (7 m) (8 m) (7 m)(8 m)cos Dˆ 39cos Dˆ 15,44 cos Dˆ 0,3 ˆD 71 1(d) Area ABCD 1 1 (1 m)(0 m) sin110 (7 m)(8 m) sin 71 05,4m substitution into cosine rule answer () substitution into sine or cosine rule answer () substitution into cosine rule cos Dˆ 0,3 answer (3) 1 (1 m)(0 m)sin110 1 (7 m)(8 m)sin 71 answer (3). b a c ac cos B b a a a a b a a cos B b a (1 cos B) b 1 cos B a b cos B 1 a cos B c a equal sides cos rule substitution simplification [10] [4] Page 5 of 6
MATHEMATICS GRADE 1 SESSION 18 (LEARNER NOTES) TOPIC : THREE-DIMENSIONAL TRIGONOMETRY 1(a) 1(b) In ADC: ˆD 65 ( s of ) AC 158 sin 65 sin 5 AC.sin 5 158.sin 65 158.sin 65 AC sin 5 AC 338,83m In ACB: BC 338,83 1500 (338,83)(1500) cos 30 BC 118, 4m In DCB: DC tanθ BC 158 tanθ 118,4 BC 1 484 499,606 θ 7,39 1(c) 1 Area ABC (338,83)(1500)sin 30 Area ABC 17061, 5m 1(d) AD (338,83) (158) AD 139769, 7689 AD 373,86m BD (118, 4) (158) BD 150946,56 BD 18, 60m (1500) (373,86) (18, 60) (373,86)(18, 60) cos ADB ˆ (373,86)(18, 60) cos ADB ˆ (373,86) (18, 60) (1500) 918648, 79 cos ADB ˆ 600770, 7404 cos ADB ˆ 0, 653971661 ADB ˆ 130,84 ˆD 65 AC 158 sin 65 sin 5 AC 338,83m (3) cosine rule to get BC BC 118,4m DC tanθ BC 158 tanθ 118,4 θ 7,39 area rule answer (5) () Pythagoras AD 373,86m BD 18,60m cosine rule substitution answer (6) [16] The SSIP is supported by Page 6 of 6