Question

The maximum value of \[ 12 \sin x - 5 \cos x + 3 \] is:

• A. 18
• B. 13
• C. 16
• D. 10

Hint:

To find the maximum of \( a \sin x + b \cos x \), use \( R = \sqrt{a^2 + b^2} \).

Answer 1

The Correct Option is C

Step 1: Expressing in \( R \sin(x + \alpha) \) Form
The given expression: \[ 12 \sin x - 5 \cos x \] is rewritten as: \[ R \sin (x + \alpha) \] where: \[ R = \sqrt{12^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \] Thus: \[ 12 \sin x - 5 \cos x = 13 \sin (x + \alpha) \] Step 2: Finding Maximum Value
Since \( \sin(x + \alpha) \) has a maximum value of 1: \[ 13 \times 1 + 3 = 16 \] Thus, the correct answer is \( 16 \).