# What Is the Cosine Function?

## What Is the Cosine Function?

The cosine function relates a given angle to the adjacent side and hypotenuse of a right triangle.

The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle.

### Dictionary Definition

The Merriam-Webster dictionary defines the cosine function as "a trigonometric function that for an acute angle is the ratio between the leg adjacent to the angle when it is considered part of a right triangle and the hypotenuse."

The cosine of an angle is given by the formula below:

In this formula, **cos** denotes the cosine function, **θ** is an angle of a right triangle, the adjacent is the length of the side adjacent the angle and the hypotenuse is the length of longest side. The image below shows what we mean:

## Interactive Widget

Use this **interactive widget** to create a right-angled triangle and then use the cosine function to calculate the hidden element. Start by selecting which element you want to hide (using the green buttons) and then clicking in the shaded area.

Angle: 0° | |

Adjacent: ? | |

Hypotenuse: 0 |

**Oops, it's broken!Turn your phone on its side to use this widget.**

## A Real Example of the Cosine Function

It is easier to understand the cosine function with an example.

### Question

Find cos 60° using the right triangle shown below.

### Step-by-Step:

# 1

Start with the formula:

cos θ = adjacent / hypotenuse

**Don't forget:** / means ÷

# 2

Substitute the angle θ, the length of the adjacent and the length of the hypotenuse into the formula. In our example, θ = 60°, the adjacent is 3 cm and the hypotenuse is 6 cm.

cos (60°) = 3 / 6

cos (60°) = 3 ÷ 6

cos (60°) = 0.5

### Answer:

cos 60° = 0.5.

## The Graph of the Cosine Function

The cosine function can be plotted on a graph.

Find the angle along the horizontal axis, then go up until you reach the cosine graph. Go across and read the value of cos θ from the vertical axis.

We can see from the graph above that cos 60° = 0.5.

## The Cosine Function and the Unit Circle

The cosine function can be related to a unit circle, which is a circle with a radius of 1 that is centered at the origin in the Cartesian coordinate system.

For a point at any angle θ, cos θ is given by the x-coordinate of the point.