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

Options

A. 3
B. 5
C. 4
D. none of these

Answer

C

Explanation

We have,
`y=1+sin ((pi)/(4)+theta)+2cos((pi)/(4)-theta)`
`impliesy=1+(1)/(sqrt2)(sin theta+costheta)+sqrt2(cos theta+sin theta)`
`impliesy=1+(sqrt2+(1)/(sqrt2))sintheta+(sqrt2+(1)/(sqrt2))costheta [becausecostheta+sinthetalesqrt2]`
`impliesy=1+(sqrt2+(1)/(sqrt2))(cos theta+sintheta)`
`impliesyle1(sqrt2+(1)/(sqrt2))sqrt2impliesyle4`
Hence, the maximum value is 4,