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 $-1 \leq \sin \left(\theta+\frac{\pi}{4}\right) \leq 1$
$\Rightarrow -\sqrt{2} \leq \sqrt{2} \sin \left(\theta+\frac{\pi}{4}\right) \leq \sqrt{2}$
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}$