When does the tangent line to the sine curve pass through the origin? I am trying to find values $a$ and $w$ for which the line $y=ax$ is tangent to the curve $y=\sin(x)$ at $x=w$.
One immediate solution is $a=1$ and $w=0$, but I would like $a<0$ and $w\in (\pi,3\pi/2)$, and I am having a hard time finding such a solution. Asking Mathematica doesn't help (Solve, Reduce, Minimize all say This system cannot be solved) and asking Google gives me elementary calculus problems. Here is a picture showing such values of $a$ and $w$ exist (and giving an impression of how the problem changes with $a$ changing):

It is clear that there is some $a\in(-.3,-.2)$ such that $y=ax$ is tangent to $\sin(x)$, and I could find a numerical approximation, but I want an exact solution.
Thought 1: Describe the tangent line to $\sin(x)$ at $(w,\sin(w))$ as $y=\sin(w)+\cos(w)(x-w)$, and find when that intersects the origin. This leads to solving the equation $\tan(w)=w$, seemingly impossible.
Thought 2: Given $a$, I can find a spot on the sine curve with slope $a$. Indeed, such a point is $\cos(w)=a$, or $w=\arccos(a)$. Due to the limited range of $\arccos$ and wanting $w\in (\pi,3\pi/2)$, I need $w=2\pi-\arccos(a)$. Now I have to solve $a(2\pi-\arccos(a))=\sin(2\pi-\arccos(a))$, which can be simplified by the periodicity of $\sin$ and knowing $\sin(\arccos(a))=\sqrt{1-a^2}$, but that doesn't help too much.
My problem is actually slightly more general, I have the curve $y=\sin(cx)$ for some changing $c$. I know there exists exactly one triple $(a,c,w)$, for $a<0$ and $w\in (\pi/c, 3\pi/2c)$ such that $y=ax$ is tangent to $y=\sin(cx)$ at $x=w$, but I am worried this is a problem that cannot be solved. I tried doing this in the (seemingly) simplest case $c=1$ above, but to no avail. I would be very glad to know if there are related problems / solutions / approaches.

well, as already pointed out, the $x$ values of the intersection points solve $x = \tan x.$ There are infinitely many such points, you are asking about the first one with $x > 0.$ The slope $a = \cos x.$

