# 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 .

Use   . Do not round this answer yet.

To calculate the angle use the inverse sin button on the calculator ( ). 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 .

It is not possible to use trigonometry to calculate the angle 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.      is a surd. Do not round this answer yet.

The length AC is cm.

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

The length OC is cm.

Use   . Do not round this answer yet.

To calculate the angle use the inverse tan button on the calculator ( ). 