## Trigonometric Identities and Special Triangles

There are some rules and formulae that are useful when problem solving using trigonometry.

### Identities

An identity is an equation which is true for all values of the variable.
They are studied in more detail in Year 13.

Tangent Identity

A useful formula is the relationship between the sine, cosine and tangent ratios.

Squared Identity

Another useful identity connecting the sine and cosine ratios.

 sin2x + cos2x = 1 See proof

Angle Identities

There are a couple of identities which connect the sines and cosines of angles less than 90°.

These can be verified on a calculator.

 sin x = cos(90 − x) cos x = sin(90 − x)

e.g.

 cos 10° = 0.9848 sin (90 − 10)° = sin 80° = 0.9848

### Special Triangles

Angles such as 30°, 45° and 60° are used frequently and the trigonometric ratios of these angles are obtained from the two triangles below and it would be useful to memorise them.

 Top triangle sine cosine tangent Half of an equilateral triangle of side length 2 units. 30° Bottom triangle 45° 1 An isosceles triangle with equal sides of length 1 unit. 60° √3