Take a standard sine graph with a period of $4$ such that the first peak is at $x = 1$. Take also the graph $y = x$. Take also a line $y = c - x$.
- What is the value of $c$ such that the graph it is part of passes through both other graphs and is perpendicular to both at their respective intersections (points $A$ and $B$ for the $\sin$ graph and $y = x$ graph respectively)?
- What is the ratio between the distance $(0, 0)$ to $(1, 1)$ and $AB$?
Thanks for any help.
Since the line $y=c-x$ is always orthogonal to $y=x$, the problem reduce to find $c$ such that the line $y=x$ passes through the points $(x_P,y_P)$ of the function $y=\sin(\frac{\pi}{2}x)$ where the slope of the tangent is $m=y'(x_P)=1$.
we have: $$y'=\frac{\pi}{2}\cos(\frac{\pi}{2}x)=1 $$ so, in the period $[0,4]$ we have the two solutions: $$x_P=\frac{2}{\pi}\cos^{-1}\left(\frac{2}{\pi}\right)$$ $$ x_{P'}=4-\frac{2}{\pi}\cos^{-1}\left(\frac{2}{\pi}\right) $$ as you can see in the figure.
From these we can find the value of $c$ for the lines $y=c-x$ that passes through these points (the red lines in the figure).