# 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.

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 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 $$x$$.

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.

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.

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 $$y$$.

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.

$a^2 + b^2 = c^2$

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

$4^2 + 4^2 = AC^2$

$32 = AC^2$

$AC = \sqrt{32}$

$$\sqrt{32}$$ 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.

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 ($$\tan^{-1}$$).

$x = 46.7^\circ$