Number of values of `p` for which equation `sin^3x+1+p^3-3p sinx=0(p >0)` has a root is ________

Answer

1

Explanation

`sin^(3) x+p^(3)+1=3p sin x`
`rArr (sin x p+1) (sin^(2) x+1+p^(2)-sin x-p-p sin x)=0`
therefore, either `sin x+p+1=0rArr p rArr p=-(1+sin x)`
or `sin x =1=p`
Hence, only value of `p(p gt 0)` is possible which is given by `p=1`.