I'm asking this on the math stack exchange because it seems that the key part of this physics problem I'm asking for help on is more related to the geometry of it than the physics of it.
I'm independently going through Physics for Scientists and Engineers, 3rd Edition, by Raymond A. Serway, Chapter 3, problem 80, which, to paraphrase the problem, goes like this:
A boy scout ties a rope to his food, throws the rope over a branch, and starts walking at a constant velocity $v_0$ and the food starts going straight up. The food starts at the level of his hands and at a height $h$ above that is the branch. The distance he is away from the vertical rope at any given time is $x$. (Here's my drawing of the situation below, the book has a fancier picture of a real human and a tree, but I've stripped it down to the essentials).
The problem goes on to say:
(a) Prove that the food's velocity $v$ when the boy's position is $x$ is equal to:
$v = v_0 \cdot x \cdot (h^2 + x^2)^{-1/2}.$
(b) Prove that the food's acceleration a when the boy's position is $x$ is equal to:
$a = v_0^2 \cdot h^2 \cdot (h^2 + x^2)^{-3/2}.$
I can solve part (a) by drawing a triangle down by the boy's hands as shown and understanding that the food's velocity wraps around the branch in the direction along $l$, and $v_0$ is then the hypotenuse, so $v$ is $v_0 \sin \theta$, which $\sin \theta$ is then equivalent to $x / (h^2 + x^2)^{1/2}$, so $v$ is $v_0 \cdot x \cdot (h^2 + x^2)^{-1/2}$, which is the answer.
I can also see that a is the time derivative of $v$. I can imagine using the chain rule to say that $a = dv/dt = dv/d \theta \cdot d \theta /dx \cdot dx/dt$. I can see that $dv/d \theta = v_0 \cos \theta $ and I can see that $dx/dt$ is the constant $v_0$ that the boy is moving at. I can look at the answer I am supposed to arrive at for part (b) and what I have so far and see that $d \theta /dx$ is supposed to be $\cos \theta$, but that's what I can't see as natuarally following.
Can someone help me visualize or analytically prove why in this setup, $d \theta /dx$ is necessarily $\cos \theta$?