Right Triangle Side Naming

Side Names Determine Trigonometric Ratios

In right-triangle trigonometry, a side name tells us how that side is positioned relative to a chosen acute angle. Once the sides are identified correctly, their ratios can be used to calculate distances and heights that are difficult to measure directly.

For example, the height of a tower can be represented by one side of a right triangle. A measured horizontal distance and angle of elevation then determine which trigonometric ratio to use.

Three Names of Sides in a Right Triangle

Choose one of the two acute angles as the reference angle. If that angle is $\theta$, the sides are named as follows:

1. Opposite side: the side across from $\theta$. It does not touch the reference angle.

2. Adjacent side: the leg that touches $\theta$. The hypotenuse also touches an acute angle, so the phrase "adjacent side" specifically means the touching side that is not the hypotenuse.

3. Hypotenuse: the side opposite the right angle, $90^\circ$. It is always the longest side of a right triangle.

What Changes When the Reference Angle Changes

The hypotenuse is determined by the right angle, so its name never changes. The other two names are relative. If the reference moves from one acute angle to the other, the former opposite side becomes adjacent and the former adjacent side becomes opposite.

This check prevents a common error: identify the reference angle $\theta$ first, then label the sides, and only then choose sine, cosine, or tangent.