Final answer:
The question cannot be accurately answered without additional context or a diagram, but it appears to involve trigonometric relationships, the Pythagorean theorem, vector addition, or coordinate transformation.
Step-by-step explanation:
Without the full context or a provided diagram, it's challenging to solve for x and y or to find the measures of angles QPS and SPT. However, based on the various pieces of information given, it seems that the question involves trigonometric relationships, the Pythagorean theorem, vector addition, or coordinate transformation related to angles and distances in a geometric figure.
To provide accurate assistance, it would be necessary to have the specific equations or geometric setup, which could include details like lengths of sides, angle measures already known, or the positions of points in a coordinate system. With such information, one could apply trigonometric ratios, the Pythagorean theorem, vector analysis, or coordinate transformation techniques to solve for unknown variables and angle measures.
The question mentions using the tan function to solve for distances, referring to a figure for vector addition, and includes a formula for coordinate transformation during rotation. Without the context, this information is not directly usable to solve for x and y.