# Using the Sine Function to Find the Hypotenuse

## 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.

### 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

This is rearranged to get the formula at the top of the page (see **Note**).

**H**ypotenuse = **O**pposite / **S**in θ