Question

If \(3\sin\theta+5\cos\theta=5\), then the value of \(5\sin\theta-3\cos\theta\) is equal to

• A. 5
• B. 3
• C. 4
• D. None of these

Hint:

Use identity: \((a\sin\theta+b\cos\theta)^2+(b\sin\theta-a\cos\theta)^2=a^2+b^2\). It simplifies such questions quickly.

Answer 1

The Correct Option is B

Step 1: Use identity for linear combination.
Consider:
\[ (3\sin\theta+5\cos\theta)^2+(5\sin\theta-3\cos\theta)^2 \] Step 2: Expand using \(\sin^2\theta+\cos^2\theta=1\).
\[ (3\sin\theta+5\cos\theta)^2 = 9\sin^2\theta+25\cos^2\theta+30\sin\theta\cos\theta \] \[ (5\sin\theta-3\cos\theta)^2 = 25\sin^2\theta+9\cos^2\theta-30\sin\theta\cos\theta \] Adding:
\[ = (9+25)\sin^2\theta + (25+9)\cos^2\theta =34(\sin^2\theta+\cos^2\theta)=34 \] So:
\[ (3\sin\theta+5\cos\theta)^2+(5\sin\theta-3\cos\theta)^2=34 \] Step 3: Substitute given value.
\[ 3\sin\theta+5\cos\theta=5 \Rightarrow (3\sin\theta+5\cos\theta)^2=25 \] Thus:
\[ 25+(5\sin\theta-3\cos\theta)^2=34 \Rightarrow (5\sin\theta-3\cos\theta)^2=9 \] Step 4: Take positive value as per options.
\[ 5\sin\theta-3\cos\theta = 3 \] Final Answer: \[ \boxed{3} \]