Rotating & flipping the unit circle and computing the right angle

Question

I am working in an environment, where my unit circle looks like this, rather than your typical unit circle.

I want to compute the angle of rotation with $arctan$. I cannot figure out how to transform the answer from the normal unit circle to the other coordinates, depending on the quadrant the angle is at.

For example, I know that the rotation was 204 degrees, so this result should be expected. The values for the calculation are known

$\alpha = \text{arctan}\frac{-3.1027}{6.9477}$ which comes out at $-24.07 \deg$. So how can I transform this to 204 degs? And how does the transform depend on the quadrant that the original rotation is in?

Answer 1

Assuming that those pluses and minuses indicate the orientations of the positive-sense horizontal and vertical axes, if you reflect your circle about the horizontal axis, it should be pretty obvious that the desired angle measurements are just the usual ones increased by $pi/2$ and then adjusted to land in whatever range of angles you consider valid. Use the ATAN2 function or its equivalent in your environment—i.e., the version of $\arctan$ that takes the numerator and denominator as separate arguments—to get the correct quadrant.