formula for getting the normlized X and Y values of a given degrees from a linear function

Question

I have a number which we'll call α in degrees that represent the angle of a linear function with the X axis. for example when α is 0 the linear function is on the X axis , when α is 360 the linear function is on the X axis. when α is 90 the linear function is on the Y axis. and so on. I want to get from the linear function

y = αx 

the X of y = 1, and the Y of x = 1. and I am not sure how can I do that. can anyone post a quick forumla for calc such a thing? does it have a name ? I know trigo may be needed to transfer degrees for a slope that represent the value of y for X = 1.

Answer 1

Claim:

If $\alpha$ is the inclination angle of a linear function, then the slope of the function is $\arctan \alpha$

Proof

Let $\Delta ABC$ be the slope triangle of the function, with $[AB]$ lying on the x-axis. Denote by $m$ the slope of the function and by $\alpha$ the angle of the linear function with the x-axis. Since $\angle CBA=90°$, we can tell $$\tan(\angle BAC)=\tan(\alpha)=\frac{BC}{AB}=m$$

Let then simply $$X=\frac{1}{\tan(\alpha)}$$ and $$Y=\tan(\alpha)$$