GCSE 3D Trigonometry & Pythagoras Dr J Frost (jfrost@tiffin.kingston.sch.uk) @DrFrostMaths Objectives: Determine lengths within 3D diagrams by repeated application of Pythagoras’ theorem. Determine angles within 3D diagrams by use of trigonometry. Last modified: 2nd January 2019

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RECAP :: Trigonometry & Pythagoras 𝟐𝟓 𝟐𝟒 𝒙 ? 𝒙 𝟐 +𝟐 𝟒 𝟐 =𝟐 𝟓 𝟐 𝒙= 𝟐 𝟓 𝟐 −𝟐 𝟒 𝟐 =𝟕 ? 𝒄𝒐𝒔 𝟔𝟐° = 𝟏𝟑 𝒙 𝒙= 𝟏𝟑 𝒄𝒐𝒔 𝟔𝟐° =𝟐𝟕.𝟕 62 ° 𝒙 13

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. 2 3 ? Fro Tip: Leave this as 2 rather than as a decimal. The advantage is that if we square it later, we get 2 2 =2 Using the base: 1 2 + 1 2 = 2 Using blue triangle: 2 2 + 1 2 = 3 𝜃= tan −1 1 2 =35.3 1 1 1 Your Go… Find the length of diagonal connecting two opposite corners of a unit cube. 12 𝟏𝟑 ? 4 3

Further Example 2 2 2 2 13 𝟏𝟐 5 8 6 ? ? Half the base diagonal: Hint: You may want to use Pythagoras on the square base (drawing the diagonal in) Determine the height of this pyramid. 2 ? Using Pythagoras to get the length of the diagonal across the base: 𝟐 𝟐 + 𝟐 𝟐 = 𝟖 =𝟐 𝟐 The length from the corner to the centre of the base is 𝟐 𝟐 ÷𝟐= 𝟐 Then using blue triangle: 𝒉= 𝟐 𝟐 − 𝟐 𝟐 = 𝟐 2 2 Your Go… 5 𝟏𝟐 13 Half the base diagonal: 𝟔 𝟐 + 𝟖 𝟐 𝟐 =𝟓 𝒉= 𝟏𝟑 𝟐 − 𝟓 𝟐 =𝟏𝟐 ? 8 6

Further Practice [AQA FM Mock Set 4 Paper 2] A pyramid has a square base 𝐴𝐵𝐶𝐷 of sides 6cm. Vertex 𝑉, is directly above the centre of the base, 𝑋. Work out the height 𝑉𝑋 of the pyramid. 𝑿𝑨= 𝟏 𝟐 𝟔 𝟐 + 𝟔 𝟐 =𝟑 𝟐 𝑽𝑿= 𝟏𝟎 𝟐 − 𝟑 𝟐 𝟐 = 𝟏𝟎𝟎−𝟏𝟖 = 𝟖𝟐 ?

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. 𝜃 Click for Fromanimation

Example [Edexcel IGCSE June2011-4H Q22] The diagram shows a cuboid 𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻.  𝐴𝐵 = 5cm. 𝐵𝐶 = 7cm. 𝐴𝐸 = 3cm. Determine the length 𝐴𝐺. Calculate the size of the angle between AG and the plane 𝐴𝐵𝐶𝐷. Give your answer correct to 1 decimal place. ? 𝑨𝑪= 𝟓 𝟐 + 𝟕 𝟐 = 𝟕𝟒 𝑨𝑮 ‘drops’ onto 𝑨𝑪, so we want the angle 𝑮𝑨𝑪: 𝐭𝐚𝐧 −𝟏 𝟑 𝟕𝟒 =𝟏𝟗.𝟐°

Test Your Understanding Determine the angle between the line AB and the base of the cube. [Edexcel IGCSE Jan2013-3H Q21] The diagram shows a pyramid with a horizontal rectangular base PQRS. PQ = 16 cm. QR = 10 cm. M is the midpoint of the line PR. The vertex, T, is vertically above M. MT = 15 cm. Calculate the size of the angle between TP and the base PQRS. Give your answer correct to 1 decimal place. 𝐵 2 1 1 𝐴 1 ? 𝑨𝑩= 𝟏 𝟐 + 𝟏 𝟐 = 𝟐 Angle: 𝐭𝐚𝐧 −𝟏 𝟏 𝟐 =𝟑𝟓.𝟑° 𝑻𝑷 ‘drops’ onto 𝑷𝑴. 𝑷𝑴= 𝟏𝟔 𝟐 + 𝟏𝟎 𝟐 𝟐 = 𝟖𝟗 ∠𝑻𝑷𝑴= 𝐭𝐚𝐧 −𝟏 𝟏𝟓 𝟖𝟗 =𝟓𝟕.𝟖° ?

Angles between two planes This is more of a Further Maths/Additional Maths skill, but just in case… 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? 𝜃 line of greatest slope ? 𝐵 Example: Find the angle between the plane 𝐴𝐵𝐶𝐷 and the horizontal plane. 𝜽= 𝐬𝐢𝐧 −𝟏 𝟑 𝟔 =𝟑𝟎° 6𝑐𝑚 𝐶 𝜽 3𝑐𝑚 line of greatest slope ? 𝐴 𝐷

Two Plane Example 2 2 2 2 1 𝐸 𝐷 𝐶 𝑂 ? 𝐴 𝐵 𝐸 ? Suitable diagram Find the angle between the planes 𝐵𝐶𝐸 and 𝐴𝐵𝐶𝐷. 2 2 We earlier used Pythagoras to establish the height of the pyramid. Why would it be wrong to use the angle ∠𝐸𝐶𝑂? 𝑬𝑪 is not the line of greatest slope. 𝐷 𝐶 𝑂 2 ? 𝐴 2 𝐵 𝐸 ? Suitable diagram ? Solution 2 𝐷 𝐶 𝜃= tan −1 2 1 =54.7° 𝜃 1 𝑂 From 𝐸 a ball would roll down to the midpoint of 𝐵𝐶. 𝐴 𝐵

Test Your Understanding AQA FM Jan 2013 Paper 2 Q23 The diagram shows a cuboid 𝐴𝐵𝐶𝐷𝑃𝑄𝑅𝑆 and a pyramid 𝑃𝑄𝑅𝑆𝑉. 𝑉 is directly above the centre 𝑋 of 𝐴𝐵𝐶𝐷. a) Work out the angle between the line 𝑉𝐴 and the plane 𝐴𝐵𝐶𝐷. (Reminder: ‘Drop’ 𝑉𝐴 onto the plane) 𝑨𝑿= 𝟏 𝟐 𝟔 𝟐 + 𝟏𝟎 𝟐 = 𝟑𝟒 ∠𝑽𝑨𝑿= 𝐭𝐚𝐧 −𝟏 𝟓 𝟑𝟒 =𝟒𝟎.𝟔° b) Work out the angle between the planes 𝑉𝑄𝑅 and 𝑃𝑄𝑅𝑆. 𝐭𝐚𝐧 −𝟏 𝟐 𝟓 =𝟐𝟏.𝟖° ? ?

Exercise ? ? ? ? 𝐵 𝐵 8 6 𝐶 𝐶 8 5 𝐴 8 𝐴 4 (on provided worksheet) A cube has sides 8cm. Find: a) The length 𝐴𝐵. 𝟖 𝟑 b) The angle between 𝐴𝐵 and the horizontal plane. 𝟑𝟓.𝟑° 1 2 For the following cuboid, find: a) The length 𝐴𝐵. 𝟕𝟕 =𝟖.𝟕𝟕 b) The angle between 𝐴𝐵 and the horizontal plane. 𝟒𝟑.𝟏° ? ? ? ? 𝐵 𝐵 8 6 𝐶 𝐶 8 5 𝐴 8 𝐴 4

Exercise 16 𝟖𝟕 13 24 10 ? ? ? ? 𝐴 𝐵 𝐸 𝐷 𝐶 (on provided worksheet) 3 4 A radio tower 150m tall has two support cables running 300m due East and some distance due South, anchored at 𝐴 and 𝐵. The angle of inclination to the horizontal of the latter cable is 50° as indicated. 13 𝟖𝟕 16 𝐵 𝐸 24 150𝑚 𝐷 10 𝐶 𝐴 Determine the height of the pyramid. 𝟖𝟕 =𝟗.𝟑𝟑 b) Determine the angle between the line 𝐴𝐵 and the base 𝐵𝐶𝐷𝐸 of the pyramid. ∠𝑨𝑩𝑬=𝒕𝒂 𝒏 −𝟏 𝟖𝟕 𝟏𝟑 =𝟑𝟓.𝟕° 300𝑚 50° ? 𝐵 a) Find the angle between the cable attached to 𝐴 and the horizontal plane. 𝐭𝐚𝐧 −𝟏 𝟏𝟓𝟎 𝟑𝟎𝟎 =𝟐𝟔.𝟔° b) Find the distance between 𝐴 and 𝐵. 325.3m ? ? ?

Exercise ? ? (on provided worksheet) [AQA FM Mock Set 1 Paper 2] The diagram shows a vertical mast, 𝐴𝐵, 12 metres high. Points 𝐵, 𝐶, 𝐷 are on a horizontal plane. Point 𝐶 is due West of 𝐵. Calculate 𝐶𝐷 Calculate the bearing of 𝐷 from 𝐶 to the nearest degree. 5 a) 𝑩𝑪= 𝟏𝟐 𝐭𝐚𝐧 𝟑𝟓 =𝟏𝟕.𝟏𝟑𝟕𝟕𝟕𝟔 𝑩𝑫= 𝟏𝟐 𝐭𝐚𝐧 𝟐𝟑° =𝟐𝟖.𝟐𝟕𝟎𝟐𝟐𝟖 𝑪𝑫= 𝟏𝟕.𝟏… 𝟐 + 𝟐𝟖.𝟐… 𝟐 =𝟑𝟑.𝟎𝟔𝒎 b) 𝟗𝟎°+ 𝐭𝐚𝐧 −𝟏 𝟐𝟖.𝟐𝟕 𝟏𝟕.𝟏𝟒 =𝟏𝟒𝟗° ? ?

Exercise ? ? (on provided worksheet) 6 7 [Edexcel GCSE June 2004 6H Q18] The diagram represents a prism. 𝐴𝐸𝐹𝐷 is a rectangle. 𝐴𝐵𝐶𝐷 is a square. 𝐸𝐵 and 𝐹𝐶 are perpendicular to plane 𝐴𝐵𝐶𝐷. 𝐴𝐵=60 cm. 𝐴𝐷=60 cm. Angle 𝐴𝐵𝐸=90°. Angle 𝐵𝐴𝐸=30°. Calculate the size of the angle that the line 𝐷𝐸 makes with the plane 𝐴𝐵𝐶𝐷. Give your answer correct to 1 decimal place. Solution: 𝟐𝟐.𝟐° [Edexcel IGCSE May2013-4H Q22] The diagram shows a triangular prism with a horizontal rectangular base ABCD. AB = 10 cm.  BC = 7 cm. M is the midpoint of AD. The vertex T is vertically above M. MT = 6 cm. Calculate the size of the angle between TB and the base ABCD. Give your answer correct to 1 decimal place. Solution: 𝟐𝟗.𝟓° ? ?

Exercise 𝐸 𝐷 𝐶 𝑀 𝑋 𝐴 𝐵 ? ? ? ? 𝐼 𝐽 20cm 𝐺 𝐻 12𝑚 𝐹 𝐸 8𝑚 𝐷 24cm 𝐶 7𝑚 𝐴 (on provided worksheet) 𝐴 𝐵 𝐶 𝐷 𝐸 𝐹 𝐺 𝐻 𝐼 𝐽 8𝑚 12𝑚 24𝑚 7𝑚 𝐸 8 9 20cm 𝐷 𝐶 𝑀 𝑋 24cm 𝐴 24cm 𝐵 A school buys a set of new ‘extra comfort’ chairs with its seats pyramid in shape. 𝑋 is at the centre of the base of the pyramid, and 𝑀 is the midpoint of 𝐵𝐶. By considering the triangle 𝐸𝐵𝐶, find the length 𝐸𝑀. 16cm Hence determine the angle between the triangle 𝐸𝐵𝐶 and the plane 𝐴𝐵𝐶𝐷. 𝐜𝐨𝐬 −𝟏 𝟏𝟐 𝟏𝟔 =𝟒𝟏.𝟒° Frost Manor is as pictured, with 𝐸𝐹𝐺𝐻 horizontally level. a) Find the angle between the line 𝐴𝐺 and the plane 𝐴𝐵𝐶𝐷. 𝐭𝐚𝐧 −𝟏 𝟖 𝟐𝟓 =𝟏𝟕.𝟕° b) Find the angle between the planes 𝐹𝐺𝐼𝐽 and 𝐸𝐹𝐺𝐻. 𝐭𝐚𝐧 −𝟏 𝟒 𝟏𝟐 =𝟏𝟖.𝟒° ? ? ? ?

Exercise ? ? ? ? (on provided worksheet) [AQA FM June 2013 Paper 2] 𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻 is a cuboid. 𝑀 is the midpoint of 𝐻𝐺. 𝑁 is the midpoint of 𝐷𝐶. Show that 𝐵𝑁= 7.5m Work out the angle between the line 𝑀𝐵 and the plane 𝐴𝐵𝐶𝐷. 𝒕𝒂𝒏 −𝟏 𝟑 𝟕.𝟓 =𝟐𝟏.𝟖° Work out the obtuse angle between the planes 𝑀𝑁𝐵 and 𝐶𝐷𝐻𝐺. 𝟏𝟖𝟎− 𝐭𝐚𝐧 −𝟏 𝟒.𝟓 𝟔 =𝟏𝟒𝟑.𝟏° 10 [AQA FM Set 3 Paper 2] The diagram shows part of a skate ramp, modelled as a triangular prism. 𝐴𝐵𝐶𝐷 represents horizontal ground. The vertical rise of the ramp, 𝐶𝐹, is 7 feet. The distance 𝐵𝐶=24 feet. 11 ? ? You are given that 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡= 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑟𝑖𝑠𝑒 ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 a) The gradient of 𝐵𝐹 is twice the gradient of 𝐴𝐹. Write down the distance 𝐴𝐶. 48 b) Greg skates down the ramp along 𝐹𝐵. How much further would he travel if he had skated along 𝐹𝐴. 𝑭𝑩=𝟐𝟓 𝑨𝑭=𝟒𝟖.𝟓𝟏 𝟒𝟖.𝟓𝟏−𝟐𝟓=𝟐𝟑.𝟓𝟏 ? ?

Exercise ? ? ? 𝐼 2 2 𝐺 3 𝐻 5𝑚 𝐹 3 𝐸 2𝑚 𝐷 4 𝐶 4 4𝑚 𝐴 8𝑚 𝐵 (on provided worksheet) 12 𝐼 13 2 2 𝐺 3 𝐻 5𝑚 𝐹 𝐸 3 2𝑚 𝐷 4 𝐶 4 4𝑚 𝐴 8𝑚 𝐵 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. 𝐜𝐨𝐬 −𝟏 𝟏 𝟖 =𝟔𝟗.𝟑° a) Determine the angle between the line 𝐴𝐼 and the plane 𝐴𝐵𝐶𝐷. 𝐭𝐚𝐧 −𝟏 𝟓 𝟐 𝟓 =𝟒𝟖.𝟐° b) Determine the angle between the planes 𝐹𝐻𝐼 and 𝐸𝐹𝐺𝐻. 𝐭𝐚𝐧 −𝟏 𝟑 𝟒 =𝟑𝟔.𝟗° ? ? ?

Exercise ? (on provided worksheet) 𝐶 14 N Solution: 𝟕𝟔.𝟑° 5 4 𝐷 𝐵 3 𝑋 (Knowledge of sine/cosine rules required) [Edexcel GCSE Nov2007-6H Q25] The diagram shows a tetrahedron. AD is perpendicular to both AB and AC.  AB = 10 cm. AC  = 8 cm. AD  = 5 cm. Angle BAC = 90°. Calculate the size of angle BDC. Give your answer correct to 1 decimal place. Solution: 𝟕𝟔.𝟑° N 5 𝐷 4 𝐵 3 𝑋 𝐴 Determine the angle between the planes 𝐴𝐵𝐶 and 𝐴𝐵𝐷. ∠𝑫𝑨𝑩= 𝐭𝐚𝐧 −𝟏 𝟒 𝟑 → 𝑫𝑿=𝟑 𝐬𝐢𝐧 𝐭𝐚𝐧 −𝟏 𝟒 𝟑 = 𝟏𝟐 𝟓 ∠𝑪𝑿𝑫= 𝐭𝐚𝐧 −𝟏 𝟓÷ 𝟏𝟐 𝟓 =𝟔𝟒.𝟒° ?