Question

What is the maximum value of the function sin x + cos x?

Answer 1

$\sin( \alpha ) + \cos( \alpha ) \\ \\ \sin( \alpha ) + \sin(90 - \alpha ) \\ \\ 2 \: \frac{ \sin( \alpha+ 90 - \alpha) }{2} . \frac{ \cos(\alpha - 90 + \alpha) }{2} \\ \\ 2. \sin(45) . \cos(\alpha - 45) \\ \\ \sqrt{2} . \cos(\alpha - 45) \\ \\ Maximum \: \: value \: \: of \: \cos( \beta ) = 1 \\ \\ \: \color{purple} { \: \boxed {Maximum \: \: value \: \: of \: \sin( \alpha ) + \cos( \alpha ) = \sqrt{2} }}$