Question

For an acute angle $\theta$, $\sin$ $\theta$ $+$ $\cos$ $\theta$ takes the greatest value when $\theta$ is

• A. $45$ ${}^o$
• B. $46$ ${}^o$
• C. $60$ ${}^o$
• D. $90$ ${}^o$

Answer 1

Answer: (A)

$45$ ${}^o$

$\sin \theta +\cos \theta$ $=$ $\sqrt{2}(\frac{1}{\sqrt{2}}\sin \theta +\frac{1}{\sqrt{2}}\cos \theta )$

$\sqrt{2}(\cos \frac{\pi }{4}.\sin \theta +\sin \frac{\pi }{4}.\cos \theta )$

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

For maximum value of this expression, $\sin (\theta +\frac{\pi }{4})=1$

$\Rightarrow \theta =\frac{\pi }{4}$