# Using the Sine Function to Find the Opposite

## Using the Sine Function to Find the Opposite Side 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 opposite 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 Opposite Side of a Right Triangle

Finding the opposite side of a right triangle is easy when we know the angle and the hypotenuse.

### Question

What is the length of the opposite side of the right triangle shown below?

### Step-by-Step:

# 1

Start with the formula:

Opposite = sin θ × hypotenuse

# 2

Substitute the angle θ and the length of the hypotenuse into the formula. In our example, θ = 30° and the hypotenuse is 5 cm.

Opposite = sin (30°) × 5

Opposite = 0.5 × 5

Opposite = 2.5 cm

### Answer:

The length of the opposite side of a right triangle with an angle of 30° and a hypotenuse of 5 cm is 2.5 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 **O**pposite and we know the **H**ypotenuse.

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**).

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