Questions: Right Triangle Trigonometry Introduction

Question 1

If two right triangles both contain a 40° angle, what can you say about the ratio (opposite side) / hypotenuse in each triangle?

• A. The ratio is larger in the bigger triangle
• B. The ratio is the same in both triangles
• C. The ratio depends on the length of the hypotenuse
• D. The ratio is the same only if the triangles are congruent

Answer: (B)

All right triangles with the same acute angle are similar by AA similarity (they share the right angle and the given acute angle). Similar triangles have proportional sides, so the ratio of any two corresponding sides is constant — it depends only on the angle, not the size of the triangle. This is the foundational insight that makes trigonometry possible.

Question 2

In a right triangle, the labels 'opposite' and 'adjacent' refer to fixed sides of the triangle that do not change regardless of which angle you focus on.

• T. True
• F. False

Answer: False

The labels 'opposite' and 'adjacent' are relative to a specific acute angle. The side opposite angle A is adjacent to angle B, and vice versa. When you switch which angle you are considering, the opposite and adjacent sides swap. Only the hypotenuse (opposite the right angle) stays fixed. This confusion is one of the most common errors in setting up trig ratios.

Question 3

Why do trigonometric ratios like sin(θ) depend only on the angle θ and not on the size of the right triangle?

Model answer: Any two right triangles with the same acute angle θ are similar by AA similarity. Because similar triangles have proportional sides, the ratio of any two specified sides (like opposite to hypotenuse) is the same in both triangles. The size cancels out, leaving a ratio that depends only on θ.

This is the key conceptual leap of the topic. Trigonometry would be useless if the ratios varied with triangle size — you would need a different table for every triangle. AA similarity guarantees they don't, which is why a single value of sin(30°) = 0.5 works for every right triangle with a 30° angle, regardless of scale.