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 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

Slider

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

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)