First slide

Extreme values and Periodicity of Trigonometric functions

Question

The minimum value of $\cos 2 θ + \cos θ$ for real values of $θ$ , is

Moderate

A

-9/8
B

0
C

-2
D

none of these

Solution

$Let x = \cos 2 θ + \cos θ$
$\begin{array}{l} x = 2 \cos^{2} θ + \cos θ - 1 \\ \Rightarrow x = - 1 + 2 (\cos^{2} θ + \frac{1}{2} \cos θ) \\ \Rightarrow x = - 1 + 2 \{{(\cos θ + \frac{1}{4})}^{2} - \frac{1}{16}\} \\ \Rightarrow x = - \frac{9}{8} + 2 {(\cos θ + \frac{1}{4})}^{2} \\ \Rightarrow x \geq - \frac{9}{8} \end{array}$

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