I finished the inverse kinematics for my robot dog! While I already calculated angles for the femur and tibia (upper/lower linkages of the leg) based on (x,y), I had yet to calculate the hip angle based on a given (z,y) yet. The hip angle allows for the dog to move its leg outwards and inwards from the body, like your shoulder enabling your arm to swing left and right. However, there was now the challenge of maintaining a right angle between linkages 1 and 2! Instead of using the law of cosines, I just used the standard soh/cah/toa for right triangles, and tested my calculations in Desmos (here: https://www.desmos.com/calculator/hifeumrqwd).