23. 3D Trigonometry The Secret to Solving 3D Trigonometry Problems 3D Trigonometry is just the same as bog-standard, flat, normal trigonometryAll we need are the skills we learnt in the last two sections:1. Pythagoras2. Sin, Cos and TanThe only difference is that is a little bit harder to spot the right-angled trianglesBut once you spot them:Draw them out flatLabel your sidesFill in the information that you do knowWork out what you don’t in the usual way!And if you can do that, then you will be able to tick another pretty tricky topic off your list!Please Remember: You need a right-angled triangle to be able to use either Pythagoras or Sin, Cos and Tan… and I promise that will be the last time I say it!
3Example 1The diagram below shows a record breaking wedge of Cheddar Cheese in which rectangle PQRS is perpendicular (at 900 to) to rectangle RSTU. The distances are shown on the diagram.Calculate: (a) The distance QT (b) The angle QTRPSQ2.5 mRT4.9 m7.8 mU
4P S Q 2.5 m R T 4.9 m 7.8 m U Working out the answer (a): The first thing we need to figure out is what we are actually trying to work out!We need the line QT:PSQ2.5 mRT4.9 m7.8 mUNow, as I said, the key to this is spotting the right-angled triangles…Well, I can see a nice one: TQR.That contains the length we want, and we already know how long QR is…So now all we need to do is work out length TR…
5Working Out TR:POkay, if you look carefully, you should be able to see a right-angled triangle on the base of this wedge of cheeseIt’s the triangle TRU:SQ2.5 mRTT4.9 m7.8 mac?4.9 mURWell, we have two sides and we want to work out the Hypotenuse… This looks like a job for Pythagoras!U7.8 mb1. Label the sides2. Use the formula:3. Put in the numbers:
6P S Q 2.5 m R T 9.21… m Q 4.9 m 7.8 m c ? 2.5 m U b T 9.21… m R a Working Out TQ:PSOkay, so now we have all we need to be able to calculate TQ.Just make sure you draw the correct right-angled triangle!Q2.5 mRT9.21… mQ4.9 m7.8 mc?2.5 mUbT9.21… mROnce again, we have two sides and we want to work out the Hypotenuse… This looks like a job for Pythagoras!a1. Label the sides2. Use the formula:3. Put in the numbers:
7Working out the answer (b): PAgain, we must be sure we know what angle the question wants us to find!I have marked angle QTR on the diagramSo now we draw our right-angled triangle:SQ2.5 m9.54 mRT9.21 m4.9 m7.8 mUQHO9.54 mTo calculate the size on an angle, we must use either sin, cos or tan, which means first we must label our sides!Now, because we actually know all three lengths, we can choose! I’m going for tan!2.5 mT9.21 mRATan θ = o ÷ aTan θ = …15.20 (1dp)
8Example 2The diagram below shows a plan of a tent that I am trying to erect before the rain comes. OP is a vertical pole, and O is at the very centre of the rectangle QRST. The lengths and angles are as shown on the diagram. Calculate the height of the vertical pole OP.PQ48oORT5 m12 mS
9T ? c a 5 m R S 12 m b P ? T O 6.5 m o = Tan θ x a 7.22 m (2dp) 1. Working Out OT:TYou should be able to see that if we can work out OT, we will then have a right-angled triangle which will give us OP!So, to get started we need to use the base of the rectangle:Well, OT is half way along the line TR line, so it must be… 6.5m?ca5 mRS12 mbPHO?480TO2. Working Out OP:6.5 mAAnd now we have a right-angled triangle where we know one length (TR), and we know one angle (OTP)… so we can work out any side using a bit of sin, cos or tan!o = Tan θ x a7.22 m (2dp)