# A-level Mathematics/CIE/Pure Mathematics 1/Trigonometry

## The Trigonometric Functions[edit]

### Sine[edit]

The **sine** of an angle is defined as the ratio between the opposite and the hypotenuse. For a given angle, this ratio will always be the same, even if the triangle is scaled up or down.

### Cosine[edit]

The **cosine** of an angle is defined as the ratio between the adjacent and the hypotenuse.

### Tangent[edit]

The **tangent** of an angle is defined as the ratio between the opposite and the adjacent.

### The Unit Circle[edit]

The **unit circle** is a circle of radius 1. It can be used to provide an alternate way of looking at trigonometric functions.

In the unit circle, a right-angled triangle can be drawn with the radius as its hypotenuse. Thus, the hypotenuse is 1 and the sine and cosine functions refer to the coordinates of a point on the unit circle.

## Graphing Trigonometric Functions[edit]

A sine graph starts at , then oscillates with a *period* of and an *amplitude* of .

A cosine graph is like a sine graph in that it oscillates with a period of and an amplitude of , but it starts at

A tangent graph starts at , goes to infinity as it approaches , emerges from negative infinity after , then repeats this at . The tangent graph has a period of .

## Exact Values[edit]

It is useful to know the following exact values of trigonometric functions:

x/° | x/rad | sin x | cos x | tan x |
---|---|---|---|---|

0 | 0 | 0 | 1 | 0 |

30 | π/6 | 1/2 | √3/2 | 1/√3 |

45 | π/4 | 1/√2 | 1/√2 | 1 |

60 | π/3 | √3/2 | 1/2 | √3 |

90 | π/2 | 1 | 0 | undefined |

## Inverse Trigonometric Functions[edit]

**A Note on Notation**

Some sources may use , , and to represent the inverse trigonometric functions, but this notation is not endorsed by Cambridge International

The **inverse** trigonometric functions are functions that reverse the trigonometric functions, just like any other inverse function. The inverse trigonometric functions are: , which is the inverse of ; , which is the inverse of ; and , which is the inverse of .

## Trigonometric Identities[edit]

An **identity** is a statement that is always true, such as . A trigonometric identity, therefore, is a trigonometric statement that is always true.

It is helpful to know the following identities:

These identities can be used to prove other identities.

e.g. Prove that

## Solving Trigonometric Equations[edit]

When solving a trigonometric equation, it is important to keep the interval in mind.

e.g. Solve for .