Question

The least value of $\cos \theta +\sin \theta$ is

• A. $-\sqrt{2}$
• B. $-\sqrt{3}$
• C. $-\sqrt{5}$
• D. None of these

Answer 1

Answer: (A)

$-\sqrt{2}$

Let $z=\cos \theta +\sin \theta$.Divide and multiply by
$\sqrt{1^2+1^2}$ i.e.$\sqrt{2}$
$=\sqrt{2}(\frac{1}{\sqrt{2}}\cos \theta +\frac{1}{\sqrt{2}}\sin \theta )$
$\sqrt{2}(\sin \frac{\pi }{4}\cos \theta +\cos \frac{\pi }{4}\sin \theta )$
$=\sqrt{2}\sin (\frac{\pi }{4}+\theta )$
z is least if $\sin (\frac{\pi }{4}+\theta )$ is least i.e when $\sin (\frac{\pi }{4}+\theta )$ is $-1$
$\therefore$ least value of $\cos \theta +\sin \theta is-\sqrt{2}$