Find the maximum value of `1+sin((pi)/(4)+theta)`
`+2sin ((pi)/(4)-theta)` for all real values of `theta`.

Options

A. 1
B. 2
C. 3
D. 4

Answer

D

Explanation

we have `1+((pi)/(4)+theta)+2cos ((pi)/(4)-theta)`
`=1+(1)/sqrt(2)(cos theta+sin theta)+sqrt(2)(cos theta+sin theta)=1+((1)/sqrt(2)+sqrt(2))(cos theta+sin theta)`
`=1+((1)/sqrt(2)+sqrt(2)).sqrt(2)cos(theta-(pi)/(4))`
`therefore` maximum value `1+((1)/sqrt(2)+sqrt(2)).sqrt(2)=4`.