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Two right triangles of very different sizes each contain the same acute angle θ. How do their values of sin θ compare?

Class 10Introduction to TrigonometryTrigonometry
  • Asin θ is larger in the bigger triangle
  • Bsin θ is exactly the same in both, because the triangles are similar and their corresponding sides are in the same ratio
  • Csin θ is smaller in the bigger triangle, because the hypotenuse grows fastest
  • DThey cannot be compared unless the side lengths are given

Step-by-Step Solution

Both triangles have a right angle and the angle θ, so by the AA criterion they are similar. Similar triangles have corresponding sides in the same ratio, so opposite/hypotenuse is identical in both — this is exactly why sin θ depends only on the angle and not on the size of the triangle.
Answer: B) sin θ is exactly the same in both, because the triangles are similar and their corresponding sides are in the same ratio

Key idea: Why ratios — and not lengths

Two right triangles with the same acute angle θ are SIMILAR (Class 10 ch. 6 — AA criterion: both have a right angle and the angle θ).

Similar triangles → corresponding sides in fixed ratio.
So the ratio (opposite / hypotenuse) takes the SAME value in any right triangle with angle θ — regardless of how big the triangle is.

That's why sin θ, cos θ, tan θ are FUNCTIONS of the angle alone — they're well-defined.

This is the fundamental reason trigonometry works as a clean theory.

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