Find a point (x,y) at a distance 'd' from a given point(a,b) and an angle theta in 2-D.

Question

I am trying to find a point (x,y) that is at a given distance 'd' from a known point(a,b) at an angle theta. it would have been doable if I knew either x or y but both are unknown in this case and the angle (theta) may be clockwise or anti-clockwise in different situations. Also, the point (a,b) does not necessarily lie on the x-axis or the y-axis. this point lies on a given line at some angle from the true North.

Any hints on this problem would be really helpful. I have attached an image for better clarity.(point (c,d) is known).enter image description here

Answer 1

Here is the standard process (requiring a number of steps).

Firsr, start with a change of axes from

to

and follow the steps shown. The formula for rotation can be found in many youtube video.