I have been learning about d-q reference frame and have figured out the following:

\(
V_{s}*e^{j\omega t}=V_{s}cos(\omega t) + j V_{s}sin(\omega t)
\)

Where
\(
V_{ds} = V_{s}cos(\omega t)
\)
\(
V_{qs} = V_{s}sin(\omega t)
\)

Now if the resulting is j such that
\(
-j V_{s}*e^{j\omega t}
\)
then apperantly
\(
V_{ds} = V_{s}sin(\omega t)
\)
\(
V_{qs} = V_{s}cos(\omega t)
\)

Why does this happen and why is there a sign change?

 

It is surprisingly simple. The key is that (-j)(j)=1. When you expand the exponential into cosine and sine, the result will be -jVcos(wt)+(-j)(j)Vsin(wt). So you see? The Vsin(wt) becomes the real part and -jcos(wt) becomes imaginary part.