The three trigonometric ratios; sine, cosine and tangent are used to calculate angles and lengths in right-angled triangles. The sine and cosine rules calculate lengths and angles in any triangle.

Part of

**Trigonometry** involves calculating **angles** and **sides** in triangles.

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 ( ).

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:

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).