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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
Start with the formula:
Hypotenuse = adjacent / cos θ
Don't forget: / means ÷

Step 2
Substitute the angle θ and the length of the adjacent into the formula. In our example, θ = 60° and the adjacent is 4 cm.
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.

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 Hypotenuse and we know the Adjacent.



The two letters we are looking for are AH, which comes in the CAH in SOH CAH TOA.

This reminds us of the equation:
Cos θ = Adjacent / Hypotenuse
This is rearranged to get the formula at the top of the page (see Note).
Hypotenuse = Adjacent / Cos 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 Widget
Here is an interactive widget to help you learn about the cosine function.
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 Cos θ, the A for Adjacent and the H for Hypotenuse (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