Adjacent Side — Definition, Formula & Examples
The adjacent side is the side of a right triangle that forms one ray of a given acute angle and is not the hypotenuse. It sits right next to the angle you are looking at.
In a right triangle, for a specified acute angle , the adjacent side is the leg that, together with the hypotenuse, forms . It is distinguished from the opposite side, which does not touch , and from the hypotenuse, which is always the side across from the right angle.
How It Works
To find the adjacent side, start at the acute angle you care about. Two sides meet at that angle: one is the hypotenuse (the longest side, opposite the 90° angle) and the other is the adjacent side. The remaining side, across from your angle, is the opposite side. Which side counts as "adjacent" changes when you switch to the other acute angle in the triangle.
Worked Example
Problem: A right triangle has legs of length 3 and 4, and a hypotenuse of 5. Find cos(θ) where θ is the angle between the side of length 4 and the hypotenuse.
Identify the adjacent side: The angle θ is formed by the side of length 4 and the hypotenuse of length 5. The side of length 4 is the adjacent side because it touches θ and is not the hypotenuse.
Apply the cosine ratio: Cosine equals the adjacent side divided by the hypotenuse.
Answer:
Why It Matters
Correctly identifying the adjacent side is essential every time you set up a sine, cosine, or tangent ratio. Mixing it up with the opposite side is one of the most common sources of wrong answers in trigonometry problems, from basic right-triangle questions through physics force-component calculations.
Common Mistakes
Mistake: Confusing the adjacent side with the opposite side when the reference angle changes.
Correction: The labels "adjacent" and "opposite" depend entirely on which acute angle you choose. A side that is adjacent to one acute angle is opposite to the other. Always start by identifying your reference angle first.