2x = (90 +360k)
x = 45 + 180k

^but this isn't always correct either, how do i distinguish when k is nondivisible by 2?

>>8007940
it's a bit tricky since your original equation was basically for x from 0 to 360, but when you take 2x= sin^{-1}(1) you only get answers for 2x between 0 and 360 even though the number you want is more than that. i.e.
2x=sin^{-1}(1) implies 2x=90 OR 2x=450
and so x= 45 or x=225 but only the second one satisfies the equation you started with

>>8007951
when you square an equality you have done a non-reversible step which can add additional solutions, since a^2=b^2 does not imply a=b. so you always have to make sure when you do non-reversible steps that your final solutions after work with what you started with

>>8007940
>sinx + cosx = -sqrt(2)
√2cos(x-45°)=-√2
cos(x-45°)=-1
x-45° = 180°+360°k
x=225°+360°k

>>8007940
>2x = sin^-1(1)
Here.

sin(2x) = 1 means 2x is equal to pi mod (2pi).
which means x is equal to pi/2 mod (pi)

so the solutions are of the form x = pi/2 + n*pi, with n being an integer.

>>8007951
2x = 90 + 360(2k + 1)
x = 45 + 180(2k + 1)

[math]x^2

okay so
sin^2x - 2sinx +3 = 0
let x replace sinx
x^2 - 2x + 3 = 0
but we get complex solutions...

>>8007984
Well duh. 3+Sin^2 is between 3 and 4, while -2sin is between - 2 and +2
The result has to be non negative

>>8007995
so am i correct in guessing that the equation is written wrong and is unsolvable in its current form?

>>8008000
It has no real solution, yes