Solutions to $(x-x_0)\cos(x_0)+\sin(x_0)-\sin(x)=0$

Question

A tangent line to the sine function at the point $\{x_0, \sin(x_0)\}$, will intersect the sine at the points where $$(x-x_0)\cos(x_0)+\sin(x_0)-\sin(x)=0$$ The solutions to that equation looks like this:

There's the obvious $x_0=x$ solution, and all the other solutions are the same function, only shifted along the $x_0=x$ line. I'm interested in these solutions, but since they're all the same function but shifted, I'm focusing on the middle one.

If I plot just the middle one:

I've tried finding it analytically, to no avail, and functions I thought looked similar (for example $\mathrm{Si}(x)$) are not solutions to the equation.

Can anyone help me to be able to plot this function without just doing Newton steps to solve it?

I'm interested in the solution for all values of $x_0$ in the range $\left[-\frac{1}{2}\pi; \frac{1}{2}\pi\right]$ and all $x\in \mathbb{R}$ (the same as my second plot)

Answer 1

Let $$P(x_0,\sin(x_0))$$ then the equation of the tangent line is given by $$y=\cos(x_0)(x-x_0)+\sin(x_0)$$ so we get $$\sin(x)=\cos(x_0)(x-x_0)+\sin(x_0)$$ Substituting $x=a$ we get $$\sin(a)=\cos(x_0)(a-x_0)+\sin(x_0)$$ there is a difference to your equation, you have $$(a-x)\cos(a)+\sin(x)-\sin(a)=0$$?

Answer 2

It was too long for a comment, so I post it as an answer.: Let try this question as a simple case: from point (0,0) which is between $-\pi/2$ and $\pi/2$ a line tangent to the curve $y=sin x$ is drawn. find the coordinates of point the line touches the curve. We can estimate the coordinates as follows: draw a line from (0, 0) to minimum point (3\pi/2, -1). the gradient of line is:

$tan (\alpha)=\frac{-1}{3\pi/2} ≈-0.2123$ and $\alpha ≈-12^o$.

the point of tangent is near $(x=3\pi/2, y=-1)$. Let $x_0=180+[(90-12)=78^o]$.

$cos (180+78)=-cos 78=-0.2079$ is the gradient of tangent line due to the formula which is near the value of [$tan (-12^o)$]. So the coordinates of point of touch a is almost $(x_a=180 +78=258 ≈44\pi/30, y_a=-0.978)$.If the point of intersection is $(x_0, y_0)$ and point of touch is $(x_a, y_a)$ we can write:

$m=\frac{y_a-y_0}{x_a-x_0}=cos(x_a)$

This is a sort of Diophantine equation.I took minimum point $(3\pi/2, -1)$ for rough estimation of tangent coordinates.