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Trigonometry in 3 dimensions - Higher

The trigonometric ratios can be used to solve problems which involve calculating a length or an angle in a right-angled triangle.

It may be necessary to use Pythagoras' theorem and trigonometry to solve a problem.

Example

The shape ABCDEFGH is a cuboid.

Cuboid (ABCDEFGH) measuring 2cm x 3cm x 6cm

AB is 6 cm, BG is 3 cm and FG is 2 cm.

The diagonal AF is 7 cm.

Calculate the angle between AF and the ABCD. Give the answer to 3 significant figures.

The plane ABCD is the base of the cuboid. The line FC and the plane ABCD form a right angle.

Draw the right-angled triangle AFC and label the sides. The angle between AF and the plane is \(x\).

Triangle (ACF) with unknown angle, x and side, a

Use \(\sin{x} = \frac{o}{h}\)

\(\sin{x} = \frac{3}{7}\)

\(\sin{x} = 0.428571 \dotsc\). Do not round this answer yet.

To calculate the angle use the inverse sin button on the calculator (\(\sin^{-1}\)).

\(x = 25.4^\circ\)

Question

The shape ABCDV is a square-based . O is the midpoint of the square base ABCD.

Pyramid (ABCDV) with height 3cm

Lengths AD, DC, BC and AB are all 4 cm.

The height of the pyramid (OV) is 3 cm.

Calculate the angle between VC and the plane ABCD. Give the answer to 3 significant figures.