I have an axis in 3D space defined by the angle it makes with the ground plane.
Another axis passes through this axis at right angles and rotates around it over time. What angle will this second axis make with the ground plane?
I need the formula in terms of the first axis' incident angle and the rotation angle of the second axis. Is that possible?
Ideally, I'd like to plot this value like a sine wave.
If angle $\alpha$ is the angle between the first axis $a$ and the ground plane, angle $\theta$ is the angle of rotation of the second line $b$ around the first line $a$ with respect to the intersection line of the plane perpendicular to $a$ and the ground plane, and $\beta$ is the angle between the rotating line $b$ and the ground plane, then $$\sin(\beta) = \sin\Big(\frac{\pi}{2} - \alpha\Big)\,\sin(\theta) = \cos(\alpha)\sin(\theta)$$
This formula can be derived by looking at the configuration between the intersection line of the plane perpendicular to $a$ and the ground plane, the line $b$ and its orthogonal projection onto the ground plane. One can either construct the corresposnding tetrahedron with certain right angles, or one can construct a sphere centered of the common point of intersection of the ground plane, line $a$ and $b$, then look at the right-angled spherical triangle defined by the intersection points of the sphere with (i) the intersection line of the ground plane and the plane perpendicular to $a$, (ii) the line $b$ (iii) the line which is the orthogonal projection of $b$ onto the ground plane. Then the law of sines of spherical geometry is exactly the formula given above.