If `x+y=2pi//3` and `sin x//sin y=2`, then the

Options

A. number of values of `x in [0, 4pi]` are 4
B. number of values of `x in [0, 4pi]` are 2
C. number of values of `y in [0, 4pi]` are 4
D. number of values of `y in [0, 4pi]` are 8

Answer

A, C

Explanation

`x+y=2pi//3` or `y=(2pi//3)-x`
`:. sin x=2 sin ((2pi)/3-x)`
`=2[(sqrt(3)/2)cos x+(1/2) sin x]`
`=sqrt(3) cos x + sin x`
`rArr cos x =0`
`rArr x=npi+pi/2, n in Z`
`rArr y=(2pi)/3- n pi-pi/2=pi/6-n pi`
Hence, for `x in [0, 4pi], x=pi//2, 3pi//2, 5pi//2, 7pi//2` and for `y in [0, 4pi], y=pi//6, 7pi//6, 13pi//6, 19pi//6`