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. `2sqrtk`
D. `2sqrt(k+2)`

Answer

B

Explanation

It is given that `alpha, beta, tamma, delta` are the smallest positive angles in ascending order of magnitude such that
`sin alpha=sinbeta=singamma=sindelta=k`(a positive quantity)
`impliesbeta=pi-alpha,gamma=2pi+alphaand delta=3pi-alpha`
`therefore4sin""(alpha)/(2)+3sin""(beta)/(2)+2sin""(gamma)/(2)+sin""(delta)/(2)`
`=4sin""(alpha)/(2)+3cos""(alpha)/(2)-2sin""(alpha)/(2)-cos""(alpha)/(2)`
`=2(cos""(alpha)/(2)+sin""(alpha)/(2))=2sqrt(1+sinalpha)=2sqrt(1+k)`