First slide

Extreme values and Periodicity of Trigonometric functions

Question

Let $P = a \cos θ - b \sin θ$ Then for all real $θ$

Easy

A

$P > \sqrt{a^{2} + b^{2}}$
B

$P < - \sqrt{a^{2} + b^{2}}$
C

$- \sqrt{a^{2} + b^{2}} \leq P \leq \sqrt{a^{2} + b^{2}}$
D

None of these

Solution

$\begin{array}{l} P = \sqrt{a^{2} + b^{2}} \{\frac{a}{\sqrt{a^{2} + b^{2}}} \cos θ + \frac{b}{\sqrt{a^{2} + b^{2}}} \sin θ\} \\ = \sqrt{a^{2} + b^{2}} \cos (θ - α) \\ (W h e r e \cos α = \frac{a}{\sqrt{a^{2} + b^{2}}}, \sin α = \frac{b}{\sqrt{a^{2} + b^{2}}}) \\ B u t - 1 \leq \cos (θ - α) \leq 1 \\ \Rightarrow - \sqrt{a^{2} + b^{2}} \leq \sqrt{a^{2} + b^{2}} \cos (θ - α) \leq \sqrt{a^{2} + b^{2}} \\ \Rightarrow - \sqrt{a^{2} + b^{2}} \leq P \leq \sqrt{a^{2} + b^{2}} \end{array}$

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