Using the Cosine Function to Find the Adjacent
Using the Cosine Function to Find the Adjacent of a Right Triangle
The length of the adjacent is given by the formula below:
In this formula, θ is an angle of a right triangle, the adjacent is the length of the side next to the angle and the hypotenuse is the length of longest side. The image below shows what we mean:
How to Use the Cosine Function to Find the Adjacent of a Right Triangle
Finding the adjacent of a right triangle is easy when we know the angle and the hypotenuse.
What is the length of the adjacent of the right triangle shown below?
Start with the formula:
Adjacent = cos θ × hypotenuse
Substitute the angle θ and the length of the hypotenuse into the formula. In our example, θ = 60° and the hypotenuse is 5 cm.
Adjacent = cos (60°) × 5
Adjacent = 0.5 × 5
Adjacent = 2.5
The length of the adjacent of a right triangle with an angle of 60° 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 SOH CAH TOA.
Looking at the example above, we are trying to find the Adjacent and we know the Hypotenuse.
The two letters we are looking for are AH, which comes in the CAH in SOH CAH TOA.
This reminds us of the equation:
Cos θ = Adjacent / Hypotenuse
This is rearranged to get the formula at the top of the page (see Note).
Adjacent = Cos θ × Hypotenuse
The slider below gives another example of finding the adjacent of a right triangle (since the angle and hypotenuse are known).Open the slider in a new tab