A very bright young student of mine who has not learned trigonometry presented me with this question she came up with.

A unit circle is placed at the origin. At any point along the circle we can draw a tangent line and calculate its slope. Then we can plot (x-coordinate, slope). It seems to make this nice curved X shape. Can we make an equation for it?

Your task is to find an equation, without trigonometry.

Note: As interesting as it was to try drawing the shape on a whiteboard, I had to fight the urge to use trig for this equation. In fact, I think it has a nice parametric form using trig, so you could also find that. Or whatever you like. We were just playing around with some math.

(In reply to re: solution by Kenny M)

Indeed I've made the slope the x coordinate, and the x coordinate of the original circle is the dependent variable and therefore is the y coordinate of this function.

Posted by Charlie on 2024-02-19 18:17:47