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# INEQUALITY

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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