Question

If \( a \cos \theta + b \sin \theta = 4 \) and \( a \sin \theta - b \cos \theta = 3 \), then the value of \( a^2 + b^2 \) is

• A. 7
• B. 16
• C. 25
• D. 36

Hint:

To solve trigonometric equations, square both sides and use trigonometric identities to simplify.

Answer 1

The Correct Option is C

Step 1: Square both equations: \[ (a \cos \theta + b \sin \theta)^2 = 16 \] \[ (a \sin \theta - b \cos \theta)^2 = 9 \] Step 2: Add the two equations: \[ a^2 \cos^2 \theta + 2ab \cos \theta \sin \theta + b^2 \sin^2 \theta + a^2 \sin^2 \theta - 2ab \cos \theta \sin \theta + b^2 \cos^2 \theta = 25 \] Step 3: Simplifying: \[ a^2 (\cos^2 \theta + \sin^2 \theta) + b^2 (\cos^2 \theta + \sin^2 \theta) = 25 \] Since \( \cos^2 \theta + \sin^2 \theta = 1 \), we have: \[ a^2 + b^2 = 25 \] Thus, the correct answer is \( \boxed{25} \).