sinx + cosx =3/2

sin(x) + cox(x)  =  3/2

Square both sides:  [ sin(x) + cos(x) ]²  =  [ 3/2 ]²

Multiply out:               sin²(x) + 2sin(x)cos(x) + cos²(x)  =  9/4

Rearranging:             sin²(x) + cos²(x) + 2sin(x)cos(x)  =  9/4

Since  sin²(x) + cos²(x)  =  1:           1 + 2sin(x)cos(x)  =  9/4

Subtracting 1 from both sides:               2sin(x)cos(x)  =  5/4

Since  sin(2x)  =  2sin(x)cos(x):                       sin(2x)  =  5/4 

But the sine value can never be greater than 1   --->     sin(2x)  can't be 5/4     --->     There is no answer.

                     --->     This is an "impossible" problem.     <---