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Using the Sine Function to Find the Hypotenuse (Mathematics Lesson)

Using the Sine Function to Find the Hypotenuse of a Right Triangle

The sine function relates a given angle to the opposite 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 opposite is the length of the side opposite the angle and the hypotenuse is the length of longest side. The image below shows what we mean:

How to Use the Sine 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 opposite.

An Example Question

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

Step 1
Start with the formula:
Hypotenuse = opposite / sin θ
Don't forget: / means ÷

Step 2
Substitute the angle θ and the length of the opposite into the formula. In our example, θ = 30° and the opposite is 4 cm.
Hypotenuse = 4 / sin (30°)
Hypotenuse = 4 ÷ sin (30°)
Hypotenuse = 4 ÷ 0.5
Hypotenuse = 8 cm
The length of the hypotenuse of a right triangle with an angle of 30° and an opposite 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 Opposite.



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

This reminds us of the equation:
Sin θ = Opposite / Hypotenuse
This is rearranged to get the formula at the top of the page (see Note).
Hypotenuse = Opposite / Sin 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 opposite are known).
Interactive Widget
Here is an interactive widget to help you learn about the sine function.
Interactive Test
  show
 


Here's a second test on finding the hypotenuse using the sine function.
Here's a third test on finding the hypotenuse using the sine function.
Note

What Is the Sine Function?

The sine function is a trigonometric function.

The sine of a given angle in a right triangle is the ratio of the length of the opposite side to the length of the hypotenuse.

The sine function is defined by the formula:



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

How to Rearrange the Sine Function Formula

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



Here, the S stands for Sin θ, the O for Opposite and the H for Hypotenuse (from the SOH in SOH CAH TOA).

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



This leaves O over S - which means O divide by S, or, Opposite ÷ Sin θ.

This tells you that:
Hypotenuse = Opposite / Sin θ