First slide

Extreme values of trigonometric functions in trigonometry

Question

If $asin x + bcos (x + θ) + bcos (x - θ) = d,$ then the minimum value of $|\cos θ|$ is equal to

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

none of these

Solution

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

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