There's a way to do this, It's hard to differentiate between quads sometime.
For example..
30 degrees and 150 degees can't be told apart by their sin values.
sin(30) = .5
sin(150) = .5
so we have to examine their cosine values.
cos(30) = 0.866
cos(150) = -0.866
Therefore we can tell them apart by their cosine inverse values.
arccos(0.866) = 30
arccos( -0.866) = 150
What we can use the sin value for is to differentiate quadrants using a comparison operator.
if YOrientation < 0 then the degree is more than 180, if the XOrientation is also less than 0 then the degree is less than 270.
Some psuedo code would be like..
float XOrientation;
float YOrientation;
float degrees;
degrees = (arccos(XOrientation) * Pi/180);
If (YOrientation < 0)
degrees = degrees + 90;
If (XOrientation > 0)
degrees = degrees + 90;
For reference, the unit circle: