Disclaimer: I attempted to answer some of it by using my own deductions. I would feedback on that. The book gives the formulas for how x arises but my problem is understanding how the formulas arose. k: 
$x = \vert OT \vert - \vert PQ \vert$ $= r\theta - rsin\theta$
where $\vert PQ \vert = rsin(\theta)$
My first problem is understanding where $\vert PQ \vert$ comes from which is really $rsin\theta$
To begin with lets look at the basic property of a ratio between two separate parallel lines, specifically a smaller line and a larger line, this is important because remember sin is really a ratio of two sides specifically the ratio of the opposite to the hypotenuse. Which is a ratio of a smaller line to a larger line:
Lets calle the length of the smaller line $S$ and the length of the larger line $L$ and the ratio $(\frac{S}{L})$, be called $rat$. In relation to the picture $S = PQ$, $L = PC$ (radius), and $rat = sin$. Using simple algebra we can see that if we want the length of $S$ then $rat = \frac{S}{L} \Rightarrow (L)rat = \frac{L}{1} \frac{S}{L} \Rightarrow (L)rat = S$ Which from the picture translates to $(PC)sin = \vert PQ \vert$ $\Rightarrow (radius,r)(sin)$ $=$ $\vert PQ\vert$ $\Rightarrow rsin(\theta) = \vert PQ \vert$
Now as for $\vert OT \vert$ I have no explanation as to how that came to equal $r\theta$. Help? A simplified explanation similar to mine would be great. As for the former, did any of that make sense or am I just rambling a bunch of incorrect nonsense that I construed together from lower math to try and justify as to how $\vert PQ \vert$ came to equal $rsin(\theta)$, when in fact that doesn't suffice to explain how the formula arose? If this does make sense then I assume this was left out of the book, because, to anyone skilled in math, all this is implied from the diagram and it is trivial to include? Sometimes I feel like an idiot when trying to grasp math beyond the computation, if I have to go through this much trouble do so, do I belong studying math?