The number of solutions of sinx+sin3x+sin5x=0 in the interval `[(pi)/(2),3(pi)/(2)]` is

Options

A. 2
B. 3
C. 4
D. 5

Answer

B

Explanation

We have,
`sinx+sin3x+sin5x=0`
`implies sinx+sin5x+sin3x=0`
`implies2sin3xcos2x+sin3x=0`
`implies sin3x(2cos2x+1)=0`
`implies sin3x=0` or `cos2x=-(1)/(2)="cos"(2pi)/(3)`
`implies3x=npi` or `2n=2npi+-(2pi)/(3)`
`implies x=(npi)/(3)` or `x=n pi+-(pi)/(3)`
`implies x=(npi)/(3)` or `x=npi+-(pi)/(3)`
But, it is givenn that `x in [(pi)/(2),(3pi)/(2)]`
`thereforex=(2pi)/(3),pi,(4pi)/(3)`
`therefore` Number of solutions is 3.