Question

The maximum value of sin(x+π/5)+ cos(x+π/5) is attained at ?

Answer 1

We have to find the value of $x$ such that $f(x)=\sin(x+\frac{\pi}{5} )+\cos(x+\frac{\pi}{5} )$.

Rewrite the given function noting that $\cos(\frac{\pi}{4} )=\sin(\frac{\pi}{4} )=\frac{1}{\sqrt{2} }$,

[tex]f(x)=\sqrt{2}[ \frac{1}{\sqrt{2} } \sin(x+\frac{\pi}{5} )+ \frac{1}{\sqrt{2} } \cos(x+\frac{\pi}{5} )]\\ f(x)=\sqrt{2}[\cos(\frac{\pi}{4} ) \sin(x+\frac{\pi}{5} )+\sin(\frac{\pi}{4} )  \cos(x+\frac{\pi}{5} )][/tex].

[tex]f(x)=\sqrt{2} \sin(x+\frac{\pi}{5}+\frac{\pi}{4} )\\ f(x)=\sqrt{2} \sin(x+\frac{9\pi}{20} )[/tex].

We can see that $f(x)=\sqrt{2} \sin(x+\frac{9\pi}{20} )$ is maximum when $\sin(x+\frac{9\pi}{20} )$ is maximum that is when

[tex]x+\frac{9\pi}{20}=\frac{\pi}{2}\\ x=\frac{\pi}{2}-\frac{9\pi}{20}\\ x=\frac{\pi}{20}[/tex].