3D Trigonometry

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 3) Then you can find the angle x by using the opposite and hypotenuse in the upper triangle NOT TO SCALE Hyp 𝑂𝑝𝑝=𝑇𝑎𝑛𝜃×𝐴𝑑𝑗 13cm 𝑆𝑖𝑛𝜃= 𝑂𝑝𝑝 𝐴𝑑𝑗 𝑂𝑝𝑝=𝑇𝑎𝑛42×8 x 49.82° 𝑂𝑝𝑝=7.20 (2𝑑𝑝) 𝑆𝑖𝑛𝜃= 9.93 13 Opp 5.93 2) Then subtract this from the hypotenuse of the lower triangle 4cm 𝑆𝑖𝑛𝜃=0.764… 42° 5.93 𝐻𝑦𝑝= 𝐴𝑑𝑗 𝐶𝑜𝑠𝜃 𝜃=49.82° 8cm 7.20 Hyp 𝐻𝑦𝑝= 12 𝐶𝑜𝑠24 Adj Opp 24° 𝐻𝑦𝑝=13.14 (2𝑑𝑝) 12cm Adj 𝑆𝑞𝑢𝑎𝑟𝑒 𝑠𝑖𝑑𝑒=13.14−7.20=5.93

3D Trigonometry We have looked at using trigonometry in lots of situations Today we will be focusing on using it in 3D problems The key to answering questions in 3D is thinking about separate parts of them in 2D 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. Calculate the length of BD Calculate the angle between BH and the base of the cuboid 8.06cm (√65)  You can find the length of BD by just using the base H G 4cm E F 7cm D C 𝑎 2 + 𝑏 2 = 𝑐 2 6cm Sub in values 4cm 7 2 + 4 2 = 𝑐 2 Add up A B 7cm 65= 𝑐 2 Square root 8.06=𝑐 (Remember that as an exact value, c = √65!)

3D Trigonometry ABCDEFGH is a cuboid, as shown. Calculate the length of BD Calculate the angle between BH and the base of the cuboid 8.06cm (√65)  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 F θ D C √65 6cm Adj √65 θ 4cm 𝑇𝑎𝑛𝜃= 𝑂𝑝𝑝 𝐴𝑑𝑗 Sub in values A B 𝑇𝑎𝑛𝜃= 6 65 7cm Calculate 𝑇𝑎𝑛𝜃=0.744… Use inverse Tan 𝜃=36.7°

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?  We need the length of the diagonal first 𝑎 2 + 𝑏 2 = 𝑐 2 Sub in values 230 2 + 230 2 = 𝑐 2 230m Add up 105800= 𝑐 2 Square root 230m 325.27=𝑐 230m 325.27m 230m

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?  Now we can draw the height on, and halve the length we just found… 139m 𝑇𝑎𝑛𝜃= 𝑂𝑝𝑝 𝐴𝑑𝑗 Opp Sub in values 𝑇𝑎𝑛𝜃= 139 162.63 139m Calculate 𝑇𝑎𝑛𝜃=0.854… 40.5° θ Use inverse Tan 230m 𝜃=40.5° 325.27m 162.63m 162.63m Adj 230m 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?  Now we can draw the height on, and halve the length we just found… 139m So you would be walking up at an angle of 40.5° 𝑎 2 + 𝑏 2 = 𝑐 2 139m Sub in values 162.63 2 + 139 2 = 𝑐 2 40.5° Add up 230m 45771= 𝑐 2 162.63m 162.63m Square root 213.9=𝑐 230m So you would have to walk a distance of 213.9m to the top!

Summary We have looked at using Trigonometry in 3D shapes We have seen how to model this using 2D diagrams 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 4cm 42° 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?