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Rotation of geographical coordinates question

I'm working on interactive software which involves manipulation of a representation of the Earth. In the attached diagram I need to determine the "latitude" of point B' in radians, knowing the rotation through the angle A-A' and the difference in "longitude" between A-B.

I know this is relatively simple stuff using vectors/matrices, but I'm hoping there is a trig formula as my experience with them is limited. I'm really stuck! Thanks in advance.