# Trigonometric ratios

Trigonometry involves calculating angles and sides in triangles.

## Labelling the sides

The three sides of a right-angled triangle have specific 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 can be found by calculating the ratio of two sides of a right-angled triangle.

A useful way to remember these is:

, and

Or: .

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

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

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

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

The length of the third side of the triangle can be calculated using Pythagoras' theorem.

Use the trigonometric ratios to calculate accurate values for the angles 30° and 60°.

A square with side lengths of 1 cm can be used to calculate accurate values for the trigonometric ratios of 45°.

Split the square into two right-angled triangles.

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

Use the trigonometric ratios to calculate accurate values for the angle 45°.

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