Question

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

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

Answer 1

The correct option is C

$5$

Explanation for the correct option:

We know that the range of $a\cos \left(\theta \right)+b\sin \left(\theta \right)$ is $\left\{-\sqrt{(a^2+b^2)},\sqrt{(a^2+b^2)}\right\}$

Thus, the maximum value of $a\cos \left(\theta \right)+b\sin \left(\theta \right)$ is $\sqrt{(a^2+b^2)}$.

Now, to find the maximum value of $3\cos \left(\theta \right)+4\sin \left(\theta \right)$,

Put, $a=3,b=4$ in $\sqrt{(a^2+b^2)}$, we get,

$$\begin{gathered} =\sqrt{(3^2+4^2)} \\ =\sqrt{9+16} \\ =\sqrt{25} \\ =5 \end{gathered}$$

Thus, the maximum value of the given expression $3\cos \left(\theta \right)+4\sin \left(\theta \right)$ is $5$.

Hence, $\mathrm{Option}\left(\mathrm{C}\right)$ is the correct answer.