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Let $a$ and $b$ be real numbers. Find the maximum value of $a \cos \theta + b \sin \theta$ in terms of $a$ and $b.$

 Mar 10, 2022
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Consider the identity 

\(\displaystyle R\cos(\theta - \alpha) \equiv R\cos\theta\cos\alpha + R\sin\theta\sin\alpha.\)

Equating coefficients with the given expression,

\(\displaystyle R\cos\alpha = a, \\ R\sin\alpha = b.\)

Solving for

 \(\displaystyle R \; \text{ and }\; \alpha,\\ R^{2}=a^{2}+b^{2} \; \text{ and }\tan\alpha = b/a,\)

so

\(\displaystyle a\cos\theta + b\sin\theta \equiv \sqrt{a^{2}+b^{2}}\cos(\theta - \alpha)\)   where \(\displaystyle \alpha = \arctan(b/a).\)

Max will be \(\sqrt{a^{2}+b^{2}} \; \text{ when } \theta = \alpha.\)

 Mar 11, 2022

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