Trigonometry Challenges: IGCSE Further Maths with Dr. J. Frost
Explore advanced trigonometry concepts in IGCSE Further Mathematics with Dr. J. Frost, covering topics like sine/cosine rules, 3D Pythagoras, and practical applications. Test your understanding with challenging exercises and examples.
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IGCSE FM Trigonometry II Dr J Frost (jfrost@tiffin.kingston.sch.uk) Objectives: (from the specification) Last modified: 21st April 2016
What makes this topic Further Maths-ey? Technically this chapter requires no new knowledge since GCSE. Harder questions to do with sine/cosine rule (e.g. proofs or algebraic/surd sides) Harder 3D Pythagoras (e.g. angles between planes and between lines and planes)
Basic Trigonometry Recap June 2012 Paper 2 Q10 You always do sin/cos/tan of the angle, not the 7 sin28 =7 ? ?. ? 7 Always check your answer looks sensible. We expect ? to be a lot longer than 7 because the angle is shallow. ? = sin28 = 14.91 ?
More Examples 1 2 ? 1 ? 4 2 + 10 ? 3 30 ? ? ? Find ?. ? a) Determine cos? ????? = ???? =? ? ? + ?? ? ? ? ? If a fraction on each side (and nothing else), cross multiply. = b) Hence determine the length ??. Using triangle ???: ? ?? ?? = ? ? = ? ? ? ? + ?? ? ? ? + ?? = ?? ? ? ? + ?? ? ?= ? ? ? ???? = ?? ?? = ? ? = ? ? =? ? + ?? ? ? ? = ? ? + ? ?
Test Your Understanding 2 June 2013 Paper 1 Q3 ??? is a right-angled triangle. ? is a point on ??. tan? =2 2 2 1 45 3 ? Find ?. ? ????? = ? ? ? ? ? a) Work out the length of ??. ? ?? ? ?= b) Work out the length of ??. Using triangle ???: ???? =?? ? ?= ?? = ? ? = ??? ?= ? ? ? = ? ? ? =? ? ? ? ? ? ???? = ? ? ? Rationalise denominator by multiplying top and bottom by 2. ?? ??? = ??, ?? = ? ? ? =? ? ?? ? ? ?? ??? = ?? ?? = ? ?
Exercise 1 3 NO CALCULATORS Show that ? is an integer. ? = ?? 4 2 Find the exact value of ?. 1 ? ? 2 3 + 4 10 + 2 3 ? 5 2 45 ? 45 30 ? 60 ? = ? 8 + 2 ? ? = ? + ? ? ? = ? + ? ? ? 4 ? 7 Work out the area of trapezium ???? 6 Determine the area of triangle ???, giving your answer in the form ? + ? 3. 5 ? 12 ? 7 ? 3 + 3 30 ? [AQA Mock Papers] a) Using ???, find tan?. =? b) Work out the length of ??=? ? ? ? ? = ? + ? ? ? ?
3D Pythagoras The key here is identify some 2D triangle floating in 3D space. You will usually need to use Pythagoras twice. Find the length of diagonal connecting two opposite corners of a unit cube. 3? Determine the height of this pyramid. 2 ? 1 2 3 2 2 1 2 2 1 We obtained the 2 by using Pythagoras on the base of the cube. 2
Test Your Understanding AQA Mock Set 4 Paper 2 AQA Mock Set 1 Paper 2 The diagram shows a vertical mast, ??, 12 metres high. Points ?,?,? are on a horizontal plane. Point ? is due West of ?. a) Calculate ?? b) Calculate the bearing of ? from ? to the nearest degree A pyramid has a square base ???? of sides 6cm. Vertex ?, is directly above the centre of the base, ?. Work out the height ?? of the pyramid. 2 1 ?? =? ??+ ??= ? ? ? ??? ? ? ? ? ?? = a) = = ??? ?? ?? ?? ?????= ??.?????? ?? ????? = ??.?????? ??.? ?+ ??.? ? = ??.??? b) ?? + ??? ? = ??? ?? = ? ?? = ?? = ??.?? ??.?? ?
Angles between lines and planes A plane is: A flat 2D surface (not necessary horizontal). ? When we want to find the angle between a line and a plane, use the drop method imagine the line is a pen which you drop onto the plane. The angle you want is between the original and dropped lines. ? Click for Bromanimation
Angles between two planes To find angle between two planes: Use line of greatest slope * on one plane, before again dropping down. * i.e. what direction would be ball roll down? ? ? ? Example: Find the angle between the plane ???? and the horizontal plane. 6?? ? 3?? ? = ??? ?? ? ? = ?? ? ? ?
Further Example ? Find the angle between the planes ??? and ????. 2 We earlier used Pythagoras to establish the height of the pyramid. Why would it be wrong to use the angle ???? ?? is not the line of greatest slope. ? 2 ? ? 2 ? 2 2 ? ? ? 2 ? Suitable diagram ? Solution ? ? 2 ? = tan 1 ? = 54.7 1 1 ? From ? a ball would roll down to the midpoint of ??. ? ?
Test Your Understanding Jan 2013 Paper 2 Q23 The diagram shows a cuboid ???????? and a pyramid ?????. ? is directly above the centre ? of ????. a) Work out the angle between the line ?? and the plane ????. (Reminder: Drop ?? onto the plane) ?? =? ? ? ??+ ???= ?? ? ??? = ??? ? = ??.? ?? b) Work out the angle between the planes ??? and ????. ??? ?? ? = ??.? ?
Exercise 2 A cube has sides 8cm. Find: a) The length ??. ? ? b) The angle between ?? and the horizontal plane. ??.? 2 A radio tower 150m tall has two support cables running 300m due East and some distance due South, anchored at ? and ?. The angle of inclination to the horizontal of the latter cable is 50 as indicated. 1 ? ? ? 8 150? ? ? 300? 50 8 ? ? a) Find the angle between the cable attached to ? and the horizontal plane. ??? ? b) Find the distance between ? and ?. 325.3m ??? ??? ? = ??.? ?
Exercise 2 ? ? 3 4 ? 20cm ? ? 12? ? ? ? 8? ? ? ? ? 24cm ? 7? ? ? ? 24cm ? 24? A school buys a set of new extra comfort chairs with its seats pyramid in shape. ? is at the centre of the base of the pyramid, and ? is the midpoint of ??. a) By considering the triangle ???, find the length ??. 16cm b) Hence determine the angle between the triangle ??? and the plane ????. ??? ? ?? Frost Manor is as pictured, with ???? horizontally level. a) Find the angle between the line ?? and the plane ????. ??? ? b) Find the angle between the planes ???? and ????. ??? ? ?? ? ?? = ??.? ? ? ? = ??.? ? ?? ? = ??.?
Exercise 2 5 [June 2013 Paper 2] ???????? is a cuboid. ? is the midpoint of ??. ? is the midpoint of ??. a) Show that ?? = 7.5m b) Work out the angle between the line ?? and the plane ????. ? ?.? c) Work out the obtuse angle between the planes ??? and ????. ??? ??? ??.? ? ? 6 [Set 3 Paper 2] The diagram shows part of a skate ramp, modelled as a triangular prism. ???? represents horizontal ground. The vertical rise of the ramp, ??, is 7 feet. The distance ?? = 24 feet. ? ??? ? = ??.? = ???.? ???????? ???? ????????? ???????? You are given that ???????? = a) The gradient of ?? is twice the gradient of ??. Write down the distance ??. 48 b) Greg skates down the ramp along ??. How much further would he travel if he had skated along ??. ?? = ?? ?? = ??.?? ? ? ??.?? ?? = ??.??
Exercise 2 7 ? 8 2 2 ? 3 5? ? ? 3 ? 2? ? 4 ? 4 4? ? ? 8? A truncated pyramid is formed by slicing off the top of a square-based pyramid, as shown. The top and bottom are two squares of sides 2 and 4 respectively and the slope height 3. Find the angle between the sloped faces with the bottom face. ? ? a) Determine the angle between the line ?? and the plane ????. ? ? ??? ? = ??.? ? ? b) Determine the angle between the planes ??? and ????. ??? ?? ? ? ??? ? = ??.? = ??.? ?
Exercise 2 ? 5 4 ? ? 3 ? ? a) ? is a point on ?? such that ?? is the line of greatest slope on the triangle ???. Determine the length of ??. By Pythagoras ?? = ??, ?? = ?? and ?? = ?. If ?? = ?, then ?? = ? ?. Then ?? can be expressed in two different ways: ?? = ?? ??= Solving: ?? =? ? b) Hence determine ??. ?? ?.??= ? c) Hence find the angle between the planes ??? and ???. ??? ? ? ?? ? ?? ??? ? = ?.??? ? = ??.?? ? ?.???
Sine/Cosine Rule Recap 65 10 8 ? ? 40 45 5 Sine rule angle, but this time angle unknown, so put as numerator. We have two angle-side pairs involved, so use sine rule. ? ?????= ? ????? ????? ? = ?.?? ? =????? ? ???? ?? ???? =??????? ? ????? ? ? = ? ? = ??.?
Sine/Cosine Rule Recap 5 ? ? 1 2.5 40 6 2 All three sides involved (and unknown side opposite known angle), so cosine rule. ??= ??+ ?? ? ? ? ????? ? = ?.?? ? Again, all three sides involved so cosine rule. ??= ??+ ?.?? ? ? ?.? ???? ? = ?.?? ????? ????? = ?.?? ? ???? =?.?? ? ? ? = ??.?
Area of Non Right-Angled Triangles Rrecap 3cm Area = 0.5 x 3 x 7 x sin(59) = 9.00cm2 ? 59 7cm Area = 1 2? ?sin ? Where C is the angle wedged between two sides a and b.
Test Your Understanding 90? 4.6 Q3 Q1 Q2 27 130 15 ? 80 11 60? 40 ? 12 ? ????????? = 286.5? a) ? = 41.37 b) ???? = 483.63 ? = 122.8 ? ? ? 5 7 Q5 Q6 Q4 7 ? 30 61 11 ? ? 20 ? 18 11 ? ? = 7.89 ???? = 17.25 ? = 147.5 ? ? ? ? = 17.79 ? = 26.67 ???? = 73.33 Your calculator will say 32.5 , but it s clearly an obtuse angle. Remember that sin ? = sin 180 ? ? ? (Hint: cosine rule is not good here as ? is not opposite known angle)
Where it gets more Further Maths-ey You will frequently encounter either algebraic or surd sides. The approach is exactly the same as before. 2? + 12= ? + 32+ 2? + 42 2 2? + 4 ? + 3 cos60 1 2 4?2+ 4? + 1 = ?2+ 6? + 9 + 4?2+ 16? + 16 2 2?2+ 10? + 12 4?2+ 4? + 1 = ?2+ 6? + 9 + 4?2+ 16? + 16 2?2 10? 12 4?2+ 4? + 1 = 3?2+ 12? + 13 ?2 8? 12 = 0 ? =8 2 (But ? = 4 2 7 would lead to side of 2? + 1 being negative) ? 64 4 1 12 = 4 2 7
Another Example Jan 2013 Paper 2 Q20 18 =1 18 =1 ?2= 62+ 122 2 6 12 cos30 ? = 7.44 ?? Use ???? =1 2??sin? 2 ? 2? sin30 2?2 ? = 6 ? Can now use cosine rule.
Test Your Understanding AQA Set 4 Paper 1 Frost Special ? 6 2? 60 ?2= ?2+ 3?2 2 ? 3? cos60 = ?2+ 9?2 3?2 = 7?2 ? = ? 7 ? ? + 3 Determine the value of ?. ?= ???+ ? + ?? ? ?? ? + ? ????? ???= ???+ ??+ ?? + ? ??? ?? ??? ? = ? ?? ? = ? ? = ? ? ? ?
Exercise 3 [June 2012 Paper 2 Q13] Work out angle ?. 1 3 ? + 1 ? 60 2? 1 ? = ???.?? ? Use the cosine rule to determine ?. ? + ??= ??+ ?? ?? ?? ?? ? ????? ? =? ? 2 Here is a triangle. Work out ?. ? ? = ?? ?
Exercise 3 4 5 3? 2 ? ? 45 30 3 2? The angle ? is obtuse. Determine ?. Given that the area of the triangle is 24cm2. Find the values of ? and ?. ???? =????? ? ? ? ? ???? = ? ? ?? ?? ????? = ?? ? ???= ?? ??= ?? ? = ? ? = ?.?? ?? ? ? = ??? ?? = ??? (Remember that ??? ? = ??? ??? ? ) ?
Exercise 3 [June 2012 Paper 1] Triangle ABC has an obtuse angle at C. Given that sin? =1 triangle ??? to show that angle ? = 60 6 4, use ???? ? ? ?= ???? =? ???? ? ?= ? ? ? ?.?? ? ? ? ? ? ? ? ? ? ? + ? =? =? ? ? ? ? = ? ? ? ? ? ? + ? ? ? ? = ??? ? = ??
Exercise 3 7 June 2013 Paper 2 Q23 ? In triangle ???, ?? bisects angle ???. Use the sine rule in triangles ??? and ??? to prove that ?? ??=?? ??.
