Solve any 3-D problem
A standard method from Trigonometry in 3-D: the steps and worked examples.
Steps
- Pull out flat triangles. Redraw each triangle you need as a 2-D sketch, with its right angles marked.
- Work from the triangle with the most information.
- Use Pythagoras or right-triangle trig where there is a right angle; the sine or cosine formula where there isn't.
- Keep full calculator values until the end.
Worked examples
Example 1: Solve a 3-D problem: a vertical pole PN stands at N on level ground. A is 20 m due south of N, and the angle of elevation of P from A is 30°. B is 15 m due east of N. Find the height of the pole, the distance AB and the angle of elevation of P from B.
pull out △PNA (right angle at N): PN = 20 tan 30° ≈ 11.547 m
pull out △ANB (right angle at N, south ⊥ east): AB = √(20² + 15²) = 25 m
pull out △PNB (right angle at N): tan θ = PN/NB = 11.547/15
θ ≈ 37.6°Keep 11.547 (not 11.5) in the calculator for the last step.
For the whole topic, read Trigonometry in 3-D: key points, common mistakes and how the DSE tests it.