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,

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

The two letters we are looking for are
This is rearranged to get the formula at the top of the page (see

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

Here, the

This leaves O
Please tell us using

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.## Question

What is the length of the hypotenuse of the right triangle shown below?## Step-by-Step:

## 1

Start with the formula:

Hypotenuse = opposite / sin θ

**Don't forget:**/ means ÷## 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

## Answer:

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 angle 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

## Interactive Widget

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

## You might also like...

trigonometryfinding the opposite using the sine functionunderstanding the cosine functionfinding the angle using the cosine function

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