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 Opposite and we know the Hypotenuse.

The two letters we are looking for are OH, which comes in the SOH in SOH CAH TOA. This reminds us of the equation:
Sin θ = Opposite / Hypotenuse
This is rearranged to get the formula at the top of the page (see Note).
Opposite = Sin θ × Hypotenuse

Lesson Slides

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). Open the slider in a new tab

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 Sin θ, the O for Opposite and the H for Hypotenuse (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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See Also

What is an angle? What is a right triangle? What is the opposite? What is the hypotenuse? Learn more about the sine function on a right triangle ( interactive widget)