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)

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