Trigonometry in 3D [H]

Trigonometry in 3D

Three-dimensional problems are solved by identifying right-angled triangles within the solid — usually a combination of a base triangle and a vertical triangle. The key skill is drawing a clear 2D diagram of the relevant right-angled triangle extracted from the 3D figure, then applying Pythagoras or trigonometry to that flat diagram.

Strategy

  1. Draw and label the 3D solid clearly.
  2. Identify the right-angled triangle containing the unknown length or angle.
  3. If necessary, first find an intermediate length (often a diagonal of the base).
  4. Redraw the relevant 2D triangle and solve using Pythagoras or SOH CAH TOA.

Common 3D Scenarios

  • Angle between a line and a plane: drop a perpendicular from the endpoint of the line to the plane; the angle is between the line and its projection on the plane.
  • Space diagonal of a cuboid: two applications of Pythagoras — first find the face diagonal, then use it with the height.
  • Angle of a pyramid's slant edge: find the horizontal distance from the apex's projection to the base vertex, then use trigonometry with the height.

Space Diagonal of a Cuboid

Cuboid with dimensions \( l \times w \times h \):

\[ \text{Space diagonal} = \sqrt{l^2 + w^2 + h^2} \]

Worked Example — Cuboid

A cuboid is 8 cm × 6 cm × 4 cm. Find the length of the space diagonal and the angle it makes with the base.

Step 1: base diagonal \( =\sqrt{8^2+6^2}=\sqrt{100}=10 \) cm.

Step 2: space diagonal \( =\sqrt{10^2+4^2}=\sqrt{116}\approx10.77 \) cm.

Step 3: angle with base \( =\tan^{-1}(4/10)\approx21.8° \).

 Key Takeaways

  • All 3D trig problems reduce to a 2D right-angled triangle — identify it first.
  • Space diagonal: \( \sqrt{l^2+w^2+h^2} \). Often requires an intermediate step (base diagonal first).
  • Angle between a line and a plane: use the projection of the line onto the plane as the adjacent side.
  • Draw clear, labelled 2D diagrams of each right-angled triangle you extract from the 3D shape.