Find where line intersects sine function Here we have sin((x+1)*(pi/2)):

Now let's say we have a point at (x=2, y=1)...

And we draw a line to the origin (x=0, y=0)...

If we create a point where the line intersects the sine function, we get a new point.
Is there a way to find out the (x,y) of this new point?


The line through $\;(2,1)\;$ and through the origin is $\;y=\frac12x\;$ , so you need to solve
$$\sin\left(\frac\pi2(x+1)\right)=\frac12x$$
This yields a transcendental equation which would hardly have a simple, nice-looking zero, but:
$$h(x)=\sin\left(\frac\pi2(x+1)\right)-\frac12x\implies\begin{cases}h(0)=1\\{}\\h(1)=-\frac12\end{cases}$$
Now apply the Mean Value Theorem (=Lagrange's Theorem) to $\;h\;$ and deduce the existence of a point like the one you're interested in. If you want the actual value of the point's coordinates you may need to repeat the above several times, or use some other approximation methods (say, Newton's)
