Question

Sin^2(x)−3sin(x) = −1...how do i find all solutions?

Answer 1

`x=2npi+(-1)^nsin^-1((3-sqrt5)/2)`

Explanation:

`sin^2x-3sinx=-1`
let `sinx=p`
then equation becomes
`p^2-3p=-1` now solve for `p`
as,
`p^2-3p=-1`
`=>p^2-3p+1=0`
from quadratic formula
`p=(3+-sqrt((-3)^2-4(1)(1)))/(2(1))`
`=>p=(3+sqrt5)/2` or `p=(3-sqrt5)/2`
`=>sinx=(3-sqrt5)/2,cancel((3+sqrt5)/2)`,second value not possible since `sinx` cannot exceed 1
`(3-sqrt5)/2` can be written as `sin(sin^-1((3-sqrt5)/2))`
`=:.sinx=sin(sin^-1((3-sqrt5)/2))`
`=>x=2npi+(-1)^nsin^-1((3-sqrt5)/2)`
this is general solution
put `n=0,1` for principal solution