Using the Sine Function to Find the Hypotenuse
Using the Sine Function to Find the 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.
What is the length of the hypotenuse of the right triangle shown below?
Start with the formula:
Hypotenuse = opposite / sin θ
Don't forget: / means ÷
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
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 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
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