How do I find this coordinate given the angle and distance from origin.

Question

Here is my problem, I know that if when the coordinates and angles are polar I can use:

$x_1 = d \cos(45^\circ) + x_0$

$y_1 = d\sin(45^\circ) + y_0$

But what if my coordinate system works like this? Up/North positive on the $y$-axis is always zero degrees with 90 degrees being positive on the $x$-axis, all angles are taken clockwise relative from zero (up/north)

Answer 1

Just do a little change of variable. If you have a point $(x,y) = r(\cos \theta, \sin \theta)$ and want the angle $\theta'$ it makes in the clockwise direction with the positive $y$-axis, just write $$ \theta' = \pi/2 - \theta + 2n\pi $$ for $n \in \mathbb{Z}$.