I found this image showing the allowable launch azimuths for Vandenberg AFB and I was wondering why a 180° launch azimuth doesn't match a 90° inclination orbit. This led me to question the equation cos(i) = sin (A) . cos (L). Is there something missing on it? Does the rotation of the earth have any impact on the inclination of an orbit reached by a 180° launch azimuth?
If the Earth did not rotate, a 180° launch azimuth would result in a 90° inclination orbit.
To compensate for the rotation, the azimuth should be bigger than 180°.
If the rocket were launched from the north or south pole, the azimuth would be exactly south or north, respectively.