# Using the Cosine Function to Find the Hypotenuse (Mathematics Lesson)

# Using the Cosine Function to Find the Hypotenuse of a Right Triangle

The cosine function relates a given angle to the adjacent side and hypotenuse of a right triangle.The length of the hypotenuse is given by the formula below:

In this formula,

**θ**is an angle of a right triangle, the adjacent is the length of the side next to the angle and the hypotenuse is the length of longest side. The image below shows what we mean:

# How to Use the Cosine Function to Find the Hypotenuse of a Right Triangle

Finding the hypotenuse of a right triangle is easy when we know the angle and the adjacent.#### An Example Question

What is the length of the hypotenuse of the right triangle shown below?Step 1

Hypotenuse = adjacent / cos θ

**Don't forget:**/ means ÷

Step 2

Hypotenuse = 4 / cos (60°)

Hypotenuse = 4 ÷ cos (60°)

Hypotenuse = 4 ÷ 0.5

Hypotenuse = 8 cm

The length of the hypotenuse of a right triangle with an angle of 30° and an adjacent of 4 cm is 8 cm.
Hypotenuse = 4 ÷ cos (60°)

Hypotenuse = 4 ÷ 0.5

Hypotenuse = 8 cm

# Remembering the Formula

Often, the hardest part of finding the unknown hypotenuse is remembering which formula to use.Whenever you have a right triangle where you know one side and one angle and have to find an unknown side...

......think trigonometry...

...............think sine, cosine or tangent...

........................think

**SOH CAH TOA**.

Looking at the example above, we are trying to find the

**H**ypotenuse and we know the

**A**djacent.

The two letters we are looking for are

**AH**, which comes in the

**CAH**in SOH

**CAH**TOA.

This reminds us of the equation:

**C**os θ =

**A**djacent /

**H**ypotenuse

**Note**).

**H**ypotenuse =

**A**djacent /

**C**os θ

# A Real Example of How to Use the Sine Function to Find the Hypotenuse of a Right Triangle

The slider below gives another example of finding the hypotenuse of a right triangle (since the angle and adjacent are known).##### Interactive Test

**show**

Here's a second test on finding the hypotenuse using the cosine function.

Here's a third test on finding the hypotenuse using the cosine function.

##### Note

# What Is the Cosine Function?

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

The cosine function is defined by the formula:

The image below shows what we mean by the given angle (labelled θ), the adjacent and the hypotenuse:

# How to Rearrange the Cosine Function Formula

A useful way to remember simple formulae is to use a small triangle, as shown below:Here, the

**C**stands for

**C**os θ, the

**A**for

**A**djacent and the

**H**for

**H**ypotenuse (from the

**CAH**in SOH

**CAH**TOA).

To find the formula for the Hypotenuse, cover up the H with your thumb:

This leaves A

**over**C - which means A

**divide by**C, or, Adjacent

**÷**Cos θ.

This tells you that:

Hypotenuse = Adjacent / Cos θ