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# Using the Sine Function to Find the Hypotenuse

(KS3, Year 8)

## The Lesson

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.## 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

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