Trigonometry in 3 dimensions - Higher

The trigonometric ratios can be used to solve 3-dimensional 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

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

The length of the diagonal AF is 7 cm.

Calculate the angle between AF and the plane 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 pyramid. 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 perpendicular height of the pyramid (OV) is 3 cm.

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

The plane ABCD is the base of the pyramid. The line VO and the plane ABCD form a right angle.

Draw the right-angled triangle OVC and label the sides. The angle between VC and the plane is y.

Triangle (VCO) with unknown angle y and height, 3cm

It is not possible to use trigonometry to calculate the angle y because the length of another side is required.

Pythagoras can be used to calculate the length OC.

Draw the right-angled triangle ACD and label the sides.

Right angle triangle (ACD) with sides 4cm x 4cm and one unknown

a^2 + b^2 = c^2

\text{CD}^2 + \text{AD}^2 = \text{AC}^2

4^2 + 4^2 = AC^2

32 = AC^2

AC = \sqrt{32}

\sqrt{32} is a surd. Do not round this answer yet.

The length AC is \sqrt{32} cm.

The point O is in the centre of the length AC so OC is half of the length AC.

The length OC is \frac{\sqrt{32}}{2} cm.

Triangle (VCO) with unknown angle y and length sq root 32/2

Use \tan{x} = \frac{o}{a}

\tan{x} = \frac{3}{\frac{\sqrt{32}}{2}}

\tan{x} = 1.06066 \dotsc. Do not round this answer yet.

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

x = 46.7^\circ