# Using the Sine Function to Find the Opposite (Mathematics Lesson)

# 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.#### An Example Question

What is the length of the opposite side of the right triangle shown below?Step 1

Opposite = sin θ × hypotenuse

Step 2

Opposite = sin (30°) × 5

Opposite = 0.5 × 5

Opposite = 2.5 cm

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.
Opposite = 0.5 × 5

Opposite = 2.5 cm

# Remembering the Formula

Often, the hardest part of finding the unknown opposite 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

# A Real Example of How to Use the Sine Function to Find the Opposite Side of a Right Triangle

The slider below gives another example of finding the opposite side of a right triangle using the sine function (since the angle and hypotenuse are known).##### Interactive Test

**show**

Here's a second test on finding the opposite using the sine function.

Here's a third test on finding the opposite using the sine function.

##### Note

# 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