Introduction to Trigonometry
RD Sharma
CBSE
Grade 10
Question:
If $a \cos \theta + b \sin \theta = m$ and $a \sin \theta - b \cos \theta = n$, prove that $a^2 + b^2 = m^2 + n^2$.
Step-by-Step Solution
Key Concept: $m^2 = a^2 \cos^2\theta + b^2 \sin^2\theta + 2ab \sin\theta \cos\theta$.<br>$n^2 = a^2 \sin^2\theta + b^2 \cos^2\theta - 2ab \sin\theta \cos\theta$.<br>Add: $m^2 + n^2 = a^2(\cos^2\theta + \sin^2\theta) + b^2(\sin^2\theta + \cos^2\theta) = a^2 + b^2$.
$m^2 = a^2\cos^2\theta + b^2\sin^2\theta + 2ab\sin\theta\cos\theta$. [1.0 Mark]
$n^2 = a^2\sin^2\theta + b^2\cos^2\theta - 2ab\sin\theta\cos\theta$. [1.0 Mark]
$m^2 + n^2 = a^2(\cos^2\theta + \sin^2\theta) + b^2(\sin^2\theta + \cos^2\theta) = a^2 + b^2$. Proved! [1.0 Mark]
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🎯 Official CBSE Marking Scheme:
Squaring $m$: 1.0 Mark
Squaring $n$: 1.0 Mark
Adding equations to get $a^2 + b^2 = m^2 + n^2$: 1.0 Mark
Correct Answer:
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