Question

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

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

Answer 1

The correct option is B $\theta ={45}^o$

Let $f\left(x\right)=\sin \theta +\cos \theta =\sqrt{2}\cdot \sin (\theta +\frac{\pi }{4})$

But $-1\le \sin (\theta +\frac{\pi }{4})\le 1$

$-\sqrt{2}\le \sqrt{2}\sin (\theta +\frac{\pi }{4})\le \sqrt{2}$

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

i.e., $\sqrt{2}\sin (\theta +\frac{\pi }{4})=\sqrt{2}$

$\therefore \sin \left(\theta +\frac{\pi }{4}\right)$ $=1\Rightarrow \sin (\theta +\frac{\pi }{4})=\sin \frac{\pi }{2}$.

$\Rightarrow \theta +\frac{\pi }{4}=\frac{\pi }{2}\Rightarrow \theta =\frac{\pi }{4}={45}^o$.