I'm a neuroscientist PhD student working on a localization problem in the brain. Rather than explain the actual details of the problem, I'll give an abstract version of it.
Imagine a clear glass marble. A small bead of red metal is embedded inside, not at the center nor at the surface. Two circles are draw around the marble at the equator and meridian. There is also a small star drawn on the marble's surface, somewhere along the meridian between the equator and upper pole.
You tilt the sphere downward 24 degrees relative to the star (the star is at the vertex of the angle formed by tilting downward). After doing so, the red bead is located at the following coordinates relative to the star, as measured by a rigid ruler: 1.5 mm away from star along a line perpendicular to the meridian; 3.5 forward from the star along a line parallel to the meridian; and then 4 mm straight down into the marble.
Knowing these coordinates, what formula would you use to determine the new coordinates of the bead, again relative to the star, if you were to tilt the marble up or down from 24 degrees? For instance, what would the coordinates be if you tilted the marble down another 21 degrees, or tilted it back up to 0 degrees?
Note that there is no azimuthal movement, only tilt, like a head nodding.
In the diagram below, the orange arrows show the measurements made outside the marble using a rigid ruler. They are perpendicular and on the plane horizontal to the ground. The 1.5 mm arrow was difficult to draw — if you had your head against the wall, the arrow would be drawn on the wall, and the 3.5 mm arrow would be perpendicular to the wall. The blue arrow points down into the marble. The view is from the side, so the meridian is the circle, and the tilt is a clockwise turn of the circle. Degrees are for the angle of the dashed lines, the upper line being parallel to the ground. The exact radius is unknown, but it is approximately 5 mm.