Trigonometry in 3-D

What you will learn

• 14.6 Projection of a point or line onto a plane.
• 14.7 The angle between a line and a plane; the angle between two planes; inclination.
• 14.8 The theorem of three perpendiculars.
• 14.9 2-D and 3-D problems: angles between two lines, a line and a plane, two planes; distances between points, from a point to a line, and from a point to a plane.

You need first: Trigonometry, Trigonometry (2-D), Geometry and data basics, Arcs, sectors and mensuration

Key terms

Standard steps

Angle between a line and a plane

1. From a point P on the line, drop a perpendicular to the plane, meeting it at N (N is the projection of P).
2. Join N to the point A where the line meets the plane.
3. The angle is ∠PAN, in the right-angled triangle PAN.

Angle between two planes

1. Find the line of intersection of the two planes.
2. In each plane, draw a line perpendicular to the line of intersection, both meeting it at the same point.
3. The angle between these two lines is the angle between the planes.

Theorem of three perpendiculars

If PN is perpendicular to a plane, and NM (in the plane) is perpendicular to a line ℓ in the plane, then PM is also perpendicular to ℓ.

Why it helps: it tells you which line in the sloping plane is perpendicular to the line of intersection, so you can find the angle between planes with a right triangle.

Solve any 3-D problem

1. Pull out flat triangles. Redraw each triangle you need as a 2-D sketch, with its right angles marked.
2. Work from the triangle with the most information.
3. Use Pythagoras or right-triangle trig where there is a right angle; the sine or cosine formula where there isn't.
4. Keep full calculator values until the end.

Worked examples

projection of G on the base is C, so the angle is ∠GAC
AC = √(8² + 6²) = 10
tan ∠GAC = GC / AC = 5/10
∠GAC ≈ 26.6°

line of intersection: AB
in the base, BC ⊥ AB
GC ⊥ base and BC ⊥ AB, so GB ⊥ AB   (three perpendiculars)
the angle is ∠GBC
tan ∠GBC = 5/6
∠GBC ≈ 39.8°

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°

Common mistakes

• Taking any angle at the point of intersection as the angle with a plane, instead of the angle with the projection.
• For two planes, using lines that are not both perpendicular to the line of intersection, or that meet it at different points.
• Assuming an angle is 90° because it "looks right" in a 3-D drawing.
• Rounding a side to 3 significant figures and then using it again.

How the DSE tests it

• Section B: a long question built on a solid (often a pyramid, prism or folded paper), with parts asking for a length, an angle between a line and a plane, then an angle between planes or "is the angle increasing?".
• Paper 2 Section B: identify the angle between a line and a plane, or between two planes, in a diagram.

Questions you might not ask in class

More Form 5 maths topics