Using the Sine Function to Find the Hypotenuse

Using the Sine Function to Find the Hypotenuse 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 hypotenuse 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 Hypotenuse of a Right Triangle

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

Question

What is the length of the hypotenuse of the right triangle shown below?

Step-by-Step:

1

Start with the formula:

Hypotenuse = opposite / sin θ

Don't forget: / means ÷

2

Substitute the angle θ and the length of the opposite into the formula. In our example, θ = 30° and the opposite is 4 cm.

Hypotenuse = 4 / sin (30°)

Hypotenuse = 4 ÷ sin (30°)

Hypotenuse = 4 ÷ 0.5

Hypotenuse = 8 cm

Answer:

The length of the hypotenuse of a right triangle with an angle of 30° and an opposite of 4 cm is 8 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 Hypotenuse and we know the Opposite.

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).

Hypotenuse = Opposite / Sin θ

Slider

The slider below gives another example of finding the hypotenuse of a right triangle (since the angle and opposite 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)