Edexcel IGCSE Math A 4.8.HF trigonometry in three dimensions

Practise 3D trigonometry questions by locating the right triangle in the solid, finding needed lengths and then calculating the required angle.

Exam points

Find the right triangle inside a cuboid, prism or pyramid before applying trigonometry.
Calculate the angle between a line and a plane using a suitable right triangle.
Use Pythagoras first when a hidden 3D length is needed before the trig step.

[Maximum number: 3]

The diagram shows a prism ABCDEFGH with a horizontal base.

Figure for Question 4.8.HF Trigonometrical methods to solve problems in three question 1 — Edexcel IGCSE Math A Higher

Diagram NOT accurately drawn

The base of the prism, EFGH, is a square of side 12 cm.
Trapezium ADEF is a cross section of the prism where AF and DE are vertical edges.
$D E=C H=10 \mathrm{~cm}$ $A D=B C=15 \mathrm{~cm}$

Work out the size of the angle between CF and the base EFGH.

Give your answer correct to one decimal place.

$FH=\sqrt{12^2+12^2}=\sqrt{288}=12\sqrt{2}=16.97\ldots$ $\tan\angle CFH=\frac{10}{16.97\ldots}$

Final answer:
$\angle CFH=30.5^\circ$

Accept $30.4^\circ$ to $30.7^\circ$ .

Marking guidance:
- M1 for finding FH.
- M1 for a correct trigonometric statement involving CFH, for example $\tan CFH=10/FH$ .
- A1 for $30.5^\circ$ within the accepted range.

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4.8.HF Trigonometrical methods to solve problems in three question 1