Question

The maximum value of \(3\cos\theta+4\sin\theta\) is

• A. \(2\)
• B. \(4\)
• C. \(5\)
• D. \(1\)

Hint:

The maximum value of \(a\cos\theta+b\sin\theta\) is \(\sqrt{a^2+b^2}\), and the minimum value is \(-\sqrt{a^2+b^2}\).

Answer 1

The Correct Option is C

We need to find the maximum value of \[ 3\cos\theta+4\sin\theta. \] The standard result is: \[ a\cos\theta+b\sin\theta \] has maximum value \[ \sqrt{a^2+b^2}. \] Here, \[ a=3 \] and \[ b=4. \] Therefore, maximum value is \[ \sqrt{3^2+4^2}. \] \[ =\sqrt{9+16}. \] \[ =\sqrt{25}. \] \[ =5. \] So, the maximum value of \[ 3\cos\theta+4\sin\theta \] is \[ 5. \] Hence, the correct answer is \(5\).