Question

The value of the expression $\sin \, \theta + \cos \, \theta$ lies between

• A. - 2 and 2 both inclusive
• B. 0 and $\sqrt{2}$. both inclusive
• C. $- \sqrt{2}$ and $\sqrt{2}$. both inclusive
• D. 0 and 2 both inclusive.

Answer 1

Answer: (C)

$\sin\theta + \cos\theta = \sqrt{2} \left(\frac{1}{\sqrt{2}} \sin\theta + \frac{1}{\sqrt{2}} \cos\theta\right) $
$= \sqrt{2} \left(\sin\theta \cos 45^{\circ} + \cos \theta \sin45^{\circ}\right)$
$ = \sqrt{2} \sin\left(\theta + 45^{\circ}\right)$
and $ -1 \le \sin\left(\theta+45^{\circ}\right) \le 1 $
$\Rightarrow - \sqrt{2} \le\sin\theta + \cos\theta \le\sqrt{2} $