Question

The value of $\sin \theta +\cos \theta$ will be greatest, when

• A. $\theta =30^{\circ }$
• B. $\theta =45^{\circ }$
• C. $\theta =60^{\circ }$
• D. $\theta =90^{\circ }$

Answer 1

Answer: (B)

Let $f(x)=\sin \theta +\cos \theta =\sqrt{2}\cdot \sin \left(\theta +\frac{\pi }{4}\right)$
But $-1\le \sin \left(\theta +\frac{\pi }{4}\right)\le 1$
$\Rightarrow -\sqrt{2}\le \sqrt{2}\sin \left(\theta +\frac{\pi }{4}\right)\le \sqrt{2}$
Hence, the maximum value of $(\sin \theta +\cos \theta )$
i.e., $\sqrt{2}\sin \left(\theta +\frac{\pi }{4}\right)=\sqrt{2}$
$\therefore \sin \left(\theta +\frac{\pi }{4}\right)=1$
$\Rightarrow \sin \left(\theta +\frac{\pi }{4}\right)=\sin \frac{\pi }{2}$
$\Rightarrow \theta +\frac{\pi }{4}=\frac{\pi }{2}$
$\Rightarrow \theta =\frac{\pi }{4}=45^{\circ }$