I have the coordinates of point $A$ and the angle $\alpha$ (starting from $y$ axis going clockwise) and the distance from $A$ to $B$ called $s$. $B$ lies on a line that is perpendicular to the angle's side. How to find the coordinates of $B$?
Note: The angle can be up to 360 degrees. I have given two examples of the points' positions. Thank you in advance.
Let $$A=(x_1,y_1) B=(x_2,y_2) $$
Let $\theta$ be the angle and $d$ be the distance.
$$x_2 = x_1 + d \cos \theta $$ $$ y_2 = y_1 + d \sin \theta$$
In your example: Let $ \omega $ be the clockwise rotation starting from the positive y-axis.
Let $d$, again, be the distance between the points.
Let $d_{oa}$ be the distance between the origin and point $A$
$$ d_{oa} = \sqrt{x_1^2 + x_2^2}$$
The angle between $A$ and $B$ will be
$$ \theta = \arcsin \frac{d}{d_{oa}} $$ $$ If \\ \omega \in [ ( 2 n + 1) \pi, 2 ( n + 1 ) )\pi ] | n \in \mathbb{Z}$$ $$ \theta = - \arcsin \frac{d}{d_{oa}} \text{ for } x_1 < x_2 $$ $$ \arcsin \frac{d}{d_{oa}} \text{ for }x_1 > x_2 $$ $$ else \\ \arcsin \frac{d}{d_{oa}}\text{ for } x_1 < x_2 $$ $$ - \arcsin \frac{d}{d_{oa}}\text{ for } x_1 > x_2 $$
The set notation after the $if$ is a precise way to describe boundaries for your angle. If your angle is between $ \pi$ and $2\pi $ your solution will be different than if your angle is between 0 and $\pi$
It will take come spatial reasoning and analyzing your graph to arrive at your final solution. Correct me if I'm wrong.