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# Using the Sine Function to Find the Opposite

(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 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

**Note**).

**O**pposite =

**S**in θ ×

**H**ypotenuse

## 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 Opposite, cover up the O with your thumb:

This leaves S

**next to**H - which means S

**times**H, or, Sin θ

**×**Hypotenuse. This tells you that:

Opposite = Sin θ × Hypotenuse

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