The maximum value of a cos x + b sin x is | vedclass

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(D) Let $f(x) = a \cos x + b \sin x$. We can write this expression as $f(x) = \sqrt{a^2 + b^2} \left( \frac{a}{\sqrt{a^2 + b^2}} \cos x + \frac{b}{\sq

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

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(D) Let $f(x) = a \cos x + b \sin x$. We can write this expression as $f(x) = \sqrt{a^2 + b^2} \left( \frac{a}{\sqrt{a^2 + b^2}} \cos x + \frac{b}{\sqrt{a^2 + b^2}} \sin x \right)$. Let $\cos \alpha = \frac{a}{\sqrt{a^2 + b^2}}$ and $\sin \alpha = \frac{b}{\sqrt{a^2 + b^2}}$. Then $f(x) = \sqrt{a^2 + b^2} (\cos \alpha \cos x + \sin \alpha \sin x) = \sqrt{a^2 + b^2} \cos(x - \alpha)$. Since the maximum value of $\cos(x - \alpha)$ is $1$,the maximum value of $f(x)$ is $\sqrt{a^2 + b^2}$.

The maximum value of $a \cos x + b \sin x$ is

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