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Starter Can you find the value of x in this diagram to 2dp? Be careful with rounding answers! 1) Start by finding the opposite side in the left triangle NOT TO SCALE Hyp πππ = ππππ × π΄ππ 13cm x 49.82° 42° 8cm Adj πππ = πππ42 × 8 5.93 4cm 5.93 24° ππππ = πππ π΄ππ ππππ = 9.93 13 πππ = 7.20 (2ππ) Opp 7.20 3) Then you can find the angle x by using the opposite and hypotenuse in the upper triangle Opp Hyp 12cm Adj 2) Then subtract this from the hypotenuse of the lower triangle π»π¦π = π΄ππ πΆππ π π»π¦π = 12 πΆππ 24 π»π¦π = 13.14 (2ππ) πππ’πππ π πππ = 13.14 β 7.20 = 5.93 ππππ = 0.764 β¦ π = 49.82° 3D Trigonometry β’ We have looked at using trigonometry in lots of situations β’ Today we will be focusing on using it in 3D problems β’ The key to answering questions in 3D is thinking about separate parts of them in 2D β’ You will find that making quick sketches as you work will help you visualise what to do! 3D Trigonometry ABCDEFGH is a cuboid, as shown. a) Calculate the length of BD 8.06cm (β65) b) Calculate the angle between BH and the base of the cuboid ο You can find the length of BD by just using the base H G 4cm E 6cm A F D 7cm C 4cm 7cm B π2 + π 2 = π 2 72 + 42 = π 2 65 = π 2 Sub in values Add up Square root 8.06 = π (Remember that as an exact value, c = β65!) 3D Trigonometry ABCDEFGH is a cuboid, as shown. a) Calculate the length of BD 8.06cm (β65) b) Calculate the angle between BH and the base of the cuboid ο To find the angle between BH and the base, draw on BH, and look to make a right angled triangle (inside the shape) H G Opp 6 E 6cm F D β65 A ΞΈ C 7cm 4cm ΞΈ B β65 πππ ππππ = π΄ππ ππππ = 6 65 ππππ = 0.744 β¦ π = 36.7° Adj Sub in values Calculate Use inverse Tan Plenary The great pyramid of Giza is a square based pyramid of side length 230m, and is 139m high. If you were to stand in one corner and walk up the pyramid to the top, what angle would you be walking up at and how far would you have to walk? ο We need the length of the diagonal first π2 + π 2 = π 2 230m 2302 + 2302 = π 2 105800 = π 2 230m 325.27m 230m 230m 325.27 = π Sub in values Add up Square root Plenary The great pyramid of Giza is a square based pyramid of side length 230m, and is 139m high. If you were to stand in one corner and walk up the pyramid to the top, what angle would you be walking up at and how far would you have to walk? ο Now we can draw the height on, and halve the length we just foundβ¦ 139m ππππ = Opp 139m 325.27m 162.63m 230m 230m 40.5° ΞΈ πππ π΄ππ 139 ππππ = 162.63 ππππ = 0.854 β¦ π = 40.5° Sub in values Calculate Use inverse Tan 162.63m Adj So you would be walking up at an angle of 40.5° Plenary The great pyramid of Giza is a square based pyramid of side length 230m, and is 139m high. If you were to stand in one corner and walk up the pyramid to the top, what angle would you be walking up at and how far would you have to walk? ο Now we can draw the height on, and halve the length we just foundβ¦ So you would be walking up at an angle of 40.5° 139m 139m 230m 162.63m 230m 40.5° 162.63m π2 + π 2 = π 2 162.632 + 1392 = π 2 45771 = π 2 Sub in values Add up Square root 213.9 = π So you would have to walk a distance of 213.9m to the top! Summary β’ We have looked at using Trigonometry in 3D shapes β’ We have seen how to model this using 2D diagrams β’ We have seen that diagrams help a lot!! Starter (printout) Can you find the value of x in this diagram to 2dp? Be careful with rounding answers! NOT TO SCALE 13cm x 42° 4cm 8cm 24° 12cm Plenary (printout) The great pyramid of Giza is a square based pyramid of side length 230m, and is 139m high. If you were to stand in one corner and walk up the pyramid to the top, what angle would you be walking up at and how far would you have to walk?
