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

The correct option is B

$\theta ={45}^{\circ }$

$Letf\left(x\right)=sin\theta +cos\theta =\sqrt{2}sin(\theta +\frac{\pi }{4})But-1\le sin(\theta +\frac{\pi }{2})\le 1\Rightarrow -\sqrt{2}\le \sqrt{2}sin(\theta +\frac{\pi }{4})\le \sqrt{2}.$

Hence the maximum value of $(sin\theta +cos\theta )$

$i.e.,of\sqrt{2}sin(\theta +\frac{\pi }{4})=\sqrt{2}.\therefore sin(\theta +\frac{\pi }{4})=1\Rightarrow sin(\theta +\frac{\pi }{4})=sin\frac{\pi }{2}\Rightarrow \theta +\frac{\pi }{4}=\frac{\pi }{2}\Rightarrow \theta =\frac{\pi }{4}={45}^{\circ }$