# Trigonometric ratios

Trigonometry involves calculating angles and sides in triangles.

## Labelling the sides

The three sides of a right-angled triangle have special names.

The hypotenuse ( ) is the longest side. It is opposite the right angle.

The opposite side ( ) is opposite the angle in question ( ).

The adjacent side ( ) is next to the angle in question ( ).

## Three trigonometric ratios

Trigonometry involves three ratios - sine, cosine and tangent which are abbreviated to , and .

The three ratios are calculated by calculating the ratio of two sides of a right-angled triangle.

A useful way to remember these is:

## Exact trigonometric ratios for 0°, 30°, 45°, 60° and 90°

The trigonometric ratios for the angles 30°, 45° and 60° can be found using two special triangles.

An equilateral triangle with side lengths of 2 cm can be used to find exact values for the trigonometric ratios of 30° and 60°.

The equilateral triangle can be split into two right-angled triangles.

Using either of these right-angled triangles, Pythagoras can be used to find the third side of the right-angled triangle.

A right-angled isosceles triangle with two sides of length 1 cm can be used to find exact values for the trigonometric ratios of 45°.

Calculate the length of the third side of the triangle using Pythagoras' theorem.

The exact trigonometric ratios for 0°, 30°, 45°, 60° and 90° are:

is undefined because and division by zero is undefined (a calculator will give an error message).