• # question_answer The value of $\sin \theta +\cos \theta$ will be greatest when [MNR 1977, 1983; RPET 1995] A) $\theta ={{30}^{o}}$ B) $\theta ={{45}^{o}}$ C) $\theta ={{60}^{o}}$ D) $\theta ={{90}^{o}}$

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