Question

The minimum value of $\cos 2\theta +\cos \theta$ for all real values of $\theta$ is

• A. $\frac{-9}{8}$
• B. $0$
• C. $-2$
• D. $-3$

Answer 1

The correct option is A $\frac{-9}{8}$

$\cos 2\theta +\cos \theta =2{\cos }^2\theta +\cos \theta -1$
$=2({\cos }^2\theta +\frac{1}{2}\cos \theta -\frac{1}{2})$

$=2({\cos }^2\theta +\frac{1}{2}\cos \theta +\frac{1}{16}-\frac{1}{16}-\frac{1}{2})$

$=2(\cos \theta +\frac{1}{2}{)}^2-\frac{9}{8}$
The minimum value will occur when the square term is $0$. Hence the correct option is A.