Question

The maximum value of $a\cos \left(x\right)+b\sin \left(x\right)$ is

• A. $a+b$
• B. $a–b$
• C. $\left|a\right|+\left|b\right|$
• D. $\sqrt{(a^2+b^2)}$

Answer 1

The correct option is D

$\sqrt{(a^2+b^2)}$

Explanation for the correct option:

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

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

For example, the maximum value of $3\cos \left(\theta \right)+4\sin \left(\theta \right)$ is given as:

Now, placing $a=3,b=4$ 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 $3\cos \left(\theta \right)+4\sin \left(\theta \right)$ is $5$.

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