Question

For an acute angle $\alpha ,(sin\alpha +cos\alpha )$ takes the greatest value when the value of $\alpha$ is equal to _____ .

• A. ${30}^0$
• B. ${45}^0$
• C. ${60}^0$
• D. ${90}^0$

Answer 1

The correct option is B ${45}^0$

$sin\alpha +cos\alpha$
Multiplying and dividing by $\sqrt{2}$

Therefore
$\sqrt{2}(\frac{cos\alpha }{\sqrt{2}}+\frac{sin\alpha }{\sqrt{2}})$

$=\sqrt{2}(sin\frac{\pi }{4}cos\alpha +cos\frac{\pi }{4}sin\alpha )$
...$(sin\frac{\pi }{4}=cos\frac{\pi }{4}=\frac{1}{\sqrt{2}})$

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

Since $0<alpha<\frac{\pi }{2}$

Therefore equation i reaches its maximum value when $sin(\frac{\pi }{4}+\alpha )=1$

which implies
$\frac{\pi }{4}+\alpha =\frac{\pi }{2}$

$\alpha =\frac{\pi }{4}={45}^0$