First slide

Extreme values and Periodicity of Trigonometric functions

Question

$If a \sin x + b \cos (x + θ) + b \cos (x - θ) = d then the minimum value of | \cos θ | =$

Moderate

A

$\frac{1}{2 |b|} \sqrt{d^{2} - a^{2}}$
B

$\frac{1}{2 |a|} \sqrt{d^{2} - a^{2}}$
C

$\frac{1}{2 |d|} \sqrt{d^{2} - a^{2}}$
D

$0$

Solution

We have
$\begin{array}{l} a \sin x + b \cos (x + θ) + b \cos (x - θ) = d \\ \Rightarrow a \sin x + 2 b \cos x \cos θ = d \\ \Rightarrow a \sin x + (2 b \cos θ) \cos x = d \\ \Rightarrow | d | \leq \sqrt{a^{2} + 4 b^{2} \cos^{2} θ} \\ \Rightarrow d^{2} \leq a^{2} + 4 b^{2} \cos^{2} θ \\ \Rightarrow \cos θ \geq \frac{1}{2 | b |} \sqrt{d^{2} - a^{2}} \end{array}$

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