Q1. [1] § 8.3 Trigonometric Ratios of Some Specific Angles § 8.5 Summary
If $\dfrac{x}{3} = 2\sin A$, $\dfrac{y}{3} = 2\cos A$, then the value of $x^2 + y^2$ is :
- A $36$
- B $9$
- C $6$
- D $18$
Previously asked in CBSE board exam
2024 30/5/1 Q2
Generated by claude-sonnet-4-6 · 2026-06-15 10:23 · grounding rag
Model Answer
(A) 36
Given: $x = 6\sin A$, $y = 6\cos A$. So $x^2 + y^2 = 36\sin^2 A + 36\cos^2 A = 36(\sin^2 A + \cos^2 A) = 36 \times 1 = \mathbf{36}$.
Source: Introduction to Trigonometry, Section 8.5 (Identity: $\sin^2 A + \cos^2 A = 1$)
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Explanation
- First simplify: $\frac{x}{3} = 2\sin A \Rightarrow x = 6\sin A$; similarly $y = 6\cos A$.
- Then $x^2 + y^2 = 36\sin^2 A + 36\cos^2 A = 36(\sin^2 A + \cos^2 A)$.
- Apply the fundamental identity $\sin^2 A + \cos^2 A = 1$ to get 36.
- The key skill tested: algebraic substitution + Pythagorean identity. Always look to form $\sin^2 A + \cos^2 A$ when you see both sin and cos of the same angle.
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