Opposite Side — Definition, Formula & Examples

The opposite side is the side of a right triangle that is directly across from a given angle. It does not form either ray of that angle.

In a right triangle, for a specified acute angle $\theta$ , the opposite side is the side that does not serve as a leg of $\theta$ — that is, the side whose endpoints are the vertex of the right angle and the vertex of the other acute angle.

How It Works

To find the opposite side, first pick the acute angle you are working with. Then look across the triangle — the side that does not touch that angle at all is the opposite side. The two sides that do form the angle are the hypotenuse (the longest side, across from the right angle) and the adjacent side. Correctly identifying opposite versus adjacent is essential for setting up sine, cosine, and tangent ratios.

Worked Example

Problem: In right triangle ABC, angle C is 90°, angle A is 37°, and side BC = 9. Identify the opposite side relative to angle A, then find sin A.

Identify the opposite side: Angle A is at vertex A. The side that does not touch vertex A is side BC. So BC is the opposite side relative to angle A.

Identify the hypotenuse: The hypotenuse is the side opposite the right angle (angle C), which is side AB.

Write the sine ratio: Sine equals opposite over hypotenuse.

\sin A = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{AB} = \frac{9}{AB}

Answer: BC = 9 is the opposite side relative to angle A, and

\sin 37° = \frac{9}{AB}

, so

AB = \frac{9}{\sin 37°} \approx 14.95

Why It Matters

Every trigonometric ratio in a right triangle — sine, cosine, and tangent — depends on correctly labeling the opposite and adjacent sides. Misidentifying them is one of the most common sources of wrong answers in geometry and physics courses that involve force components or vector resolution.

Common Mistakes

Mistake: Confusing the opposite side with the adjacent side when the reference angle changes.

Correction: The labels opposite and adjacent are relative to a specific angle. If you switch from angle A to angle B, the opposite and adjacent sides swap. Always re-identify sides when the reference angle changes.