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A student claims that for some angle θ, sin θ = . Is this possible?

Class 10Introduction to TrigonometryTrigonometry
  • AYes, if the triangle is drawn large enough
  • BYes, whenever θ is bigger than 60°
  • CNo — sin θ is opposite/hypotenuse and the hypotenuse is the longest side, so sin θ can never be more than 1
  • DNo — a trigonometric ratio must always be a whole number

Step-by-Step Solution

In a right triangle the hypotenuse is longer than either leg, so opposite/hypotenuse is at most 1. Since > 1, no angle has that sine. (Ratios certainly need not be whole numbers — sin 30° = )
Answer: C) No — sin θ is opposite/hypotenuse and the hypotenuse is the longest side, so sin θ can never be more than 1

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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