We're building a custom bar and one of the last steps is placing the bar top on. To make it unique we want to cut a piece of butchers block diagonally to fit the width needed (15.25 inches). Its a narrow "mini" bar. The butchers block is PV=34.25 inches by PR=49.5 inches. We're nerds but we're frugal. We want to cut the largest diagonal piece of the butchers block we can for the needed width. We've laid out the problem below.
$b = 15.25$
$PV = 34.25$
$PR = 49.5$
Assume also that the center of rectangle a x b is also the center of rectangle PR x PV.
$WP = ST$
$PQ = UT$
Solve for WP an PQ.
angle t = unknown but maximum possible
side a = unknown but maximum possible
There is another similar question See "Given a width, height and angle of a rectangle, and an allowed final size, determine how large or small it must be to fit into the area" but it does not quite ask or answer the same thing.