Mathematics-Monster.com
(#mm)

menu

Cosine Function
(KS3, Year 8)

hometrigonometrythe 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:

cosine function formula 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: cosine_function_image

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. cosine function example

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.

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

Lesson Slides

The slider below gives more information about the cosine function.

Interactive Widget

Here is an interactive widget to help you learn about the cosine function on a right triangle.

Trigonometry and Right Angles

The cosine function is a function in trigonometry (called a trigonometric function). The word trigonometry comes from the Greek words 'trigonon' ("triangle") and 'metron' ("measure"). Trigonometry is the branch of mathematics that studies the relationships between the sides and the angles of right triangles. When an angle is defined in a right triangle, the three sides can be defined.

right triangle sides mini
  • The side next to the angle is called the adjacent.
  • The side opposite the angle is called the opposite.
  • The longest side is called the hypotenuse.

Other Trigonometric Functions

The cosine function is only one of the trigonometric functions:

The Cosine Function and the Unit Circle

The cosine function can be related to the unit circle. Consider a point on the unit circle (x, y). It can be joined to the center by a radius of length 1. A right triangle can be formed under the radius.

sine unit circle mini The hypotenuse is the radius of 1, the adjacent side has length x and the opposite side length y.

adjacent = cos θ × hypotenuse

x = cos θ × 1

x = cos θ

author logo

This page was written by Stephen Clarke.

You might also like...

Help Us Improve Mathematics Monster

  • Do you disagree with something on this page?
  • Did you spot a typo?
Please tell us using this form.

Find Us Quicker!

  • When using a search engine (e.g., Google, Bing), you will find Mathematics Monster quicker if you add #mm to your search term.

Share This Page

share icon

If you like Grammar Monster (or this page in particular), please link to it or share it with others.

If you do, please tell us. It helps us a lot!

Create a QR Code

create QR code

Use our handy widget to create a QR code for this page...or any page.