If `alpha,beta,gamma,delta` are the smallest positive angles in ascending order of magnitude which have their sines equal to the positive quantity `k ,` then the value of `4sinalpha/2+3sinbeta/2+2singamma/2+sindelta/2` is equal to `2sqrt(1-k)` (b) `2sqrt(1+k)` `(sqrt(1-k))/2` (d) none of these

Options

A. `2sqrt(1-k)`
B. `2sqrt(1+k)`
C. `sqrt(1+k)/(2)`
D. none of these.

Answer

B

Explanation

Since `alphaltbetaltgammaltdelta` and `sin alpha=sin beta=sin gamma=sin delta=k`, we have
`beta=pi-alpha,gamma=2pi+alpha,delta=3pi-alpha`
`rArr 4sin((alpha)/(2))+3sin((beta)/(2))+2sin((7)/(2))+sin((delta)/(2))`
`=4sin((alpha)/(2))+3cos((alpha)/(2))-2sin((alpha)/(2))-cos((alpha)/(2))`
`=2sin((alpha)/(2))+2cos((alpha)/(2))=2sqrt(1+sin alpha)=2sqrt(1+k)`