How can I find the slope of a line tangent to a small circle on a sphere? I draw a circle on the earth, so that it passes through the north pole. I then begin walking around the circle, keeping track of my latitude. How do I tell what direction that I'm facing by knowing only the diameter of the circle and my latitude? It should be understood that the line tangent to the circle lies in the plane formed by the great circle that is also tangent at the same point.

Consider a spherical triangle (composed of portions of great circles), with one vertex on the North Pole, one vertex at the center of the drawn circle, and one vertex at your location

You know your latitude, so you know your co-latitude, so you know the side of the spherical triangle from the Pole to you.

You know that the drawn circle passes through the Pole, and its diameter, so you know the distance from the Pole to the center of the drawn circle, so you know the side of the spherical triangle from the Pole to the center of the drawn circle.

You know that you are one the drawn circle, so you know the side of the spherical triangle from you to the center of the drawn circle.

With side-side-side you can solve the spherical triangle for the required information...