Trigonometry in Three Dimensions

Starting point: what you already know

• Pythagoras: $c^2=a^2+b^2$ in a right-angled triangle.
• SOH CAH TOA to find sides and angles.
• Sine rule $\frac{a}{\sin A}=\frac{b}{\sin B}$ and cosine rule $c^2=a^2+b^2-2ab\cos C$ for triangles without a right angle.
• Angle of elevation: the angle you look up from the horizontal. Angle of depression: the angle you look down.
• True bearings are measured clockwise from north, written with three digits, like $060^{\circ }$.

The method

1. Draw a clear 3D sketch. Label every point with a capital letter.
2. Mark right angles. A vertical line meets any horizontal line at $90^{\circ }$.
3. Find a triangle that contains what you want and enough known lengths or angles.
4. If it needs a length you do not have, find that length first from another triangle.
5. Redraw each triangle flat, then solve it.

The rectangular box

%%FIG:box3d%%

Heights from two sightings

A common exam type: you cannot reach the base of a tower, so you measure angles of elevation from two places.

Quick summary

• Find right-angled triangles inside the 3D shape and solve them one at a time.
• Any vertical line is at right angles to every line on the ground.
• Box diagonal: $\sqrt{a^2+b^2+c^2}$.
• With two angles of elevation, write each ground distance in terms of the height, then link them with Pythagoras or the cosine rule.

Quiz