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In right triangle PQR, right-angled at Q, PQ = 9 cm, QR = 12 cm and PR = 15 cm. A student writes "cos ". What is the mistake?

Class 10Introduction to TrigonometryTrigonometry
  • AThe student used QR, the side OPPOSITE ; the correct value is cos
  • BThe student picked the wrong hypotenuse; the correct value is cos
  • CThe student actually found cot ; the correct value is cos
  • DThere is no mistake — cos is correct

Step-by-Step Solution

For angle , the adjacent side is PQ = 9 and the hypotenuse is PR = 15, so cos . The value is QR/PR, which is sin — the student used the opposite side instead of the adjacent one.
Answer: A) The student used QR, the side OPPOSITE ; the correct value is cos

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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