How do you use the quadratic formula to solve `sintheta/2=3/(sintheta+2)` for `0<=theta<360`?

Note that if `a/b=c/d` then `axxd=bxxc`

Now we have `sintheta/2=3/(sintheta+2)`, hence

`sintheta(sintheta+2)=6`

or `sin^2theta+2sintheta-6=0`

Now as per quadratic formula, if we have a quadratic equation `ax^2+bx+c=0` in `x`, then `x=(-b+-sqrt(b^2-4ac))/(2a)`

Here we have the equation `sin^2theta+2sintheta-6=0` in `sintheta` and therefore using quadratic formula

`sintheta=(-2+-sqrt(2^2-4*1*(-6)))/2` or

`=(-2+-sqrt(28))/2=-1+-sqrt7`

But `sintheta` can have values only from `-1` to `1` and both values `-1+sqrt7` and `-1-sqrt7`, are out of the range. Hence

There is no solution.

`sin theta/2 = 3/(sin theta +2)`

`sin theta(sin theta +2) = 3 *2`
`sin^2 theta + 2sin theta = 6`
`sin^2 theta + 2sin theta - 6 = 0`

`a = 1, b = 2 and c= -6`

`sin theta = (- b +-sqrt(b^2 - 4ac))/(2a)`

`sin theta = (- 2 +-sqrt(2^2 - 4(1)(-6)))/(2(1))`

`sin theta = (- 2 +-sqrt(4 +24))/(2)`

`sin theta = (- 2 +-sqrt(28))/(2) = (- 2 +-sqrt(4*7))/(2)`

`sin theta = (- 2 +-2sqrt(7))/(2)= -1 +-sqrt(7) = -1 +- 2.656`

when `sin theta = -1 + 2.656 = 1.656,`
`theta = sin^-1 (1.656)` no solution/not valid

when `sin theta = -1 - 2.656 = -3.656,`
`theta = sin^-1 (-3.656)` no solution/not valid