Question

$ \cos 2\theta +2\cos \theta $ is always

• A. greater than $ -\frac{3}{2} $
• B. less than or equal to $ \frac{3}{2} $
• C. greater than or equal to $ \frac{-3}{2} $ and less than or equal to $ 3 $
• D. None of the above

Answer 1

Answer: (C)

We have, $ \cos 2\theta +2\cos \theta =2{{\cos }^{2}}\theta -1+2\cos \theta $
$ =2{{\left( \cos \theta +\frac{1}{2} \right)}^{2}}-\frac{3}{2} $
Now, $ 2{{\left( \cos \theta +\frac{1}{2} \right)}^{2}}\ge 0 $ for all $ \theta $
$ \therefore $ $ 2{{\left( \cos \theta +\frac{1}{2} \right)}^{2}}-\frac{3}{2}\ge \frac{-3}{2} $ for all $ \theta $
$ \Rightarrow $ $ \cos 2\theta +2\cos \theta \ge \frac{-3}{2} $ for all $ \theta $
Also, maximum value of this expression is 3.