# 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

Hypotenuse = opposite / sin θ

**Don't forget:**/ means ÷

Step 2

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.
Hypotenuse = 4 ÷ sin (30°)

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

**O**pposite.

The two letters we are looking for are

**OH**, which comes in the

**SOH**in

**SOH**CAH TOA.

This reminds us of the equation:

**S**in θ =

**O**pposite /

**H**ypotenuse

**Note**).

**H**ypotenuse =

**O**pposite /

**S**in 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).##### Quick 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

**S**in θ, the

**O**for

**O**pposite and the

**H**for

**H**ypotenuse (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