TITLE: Solving for the center of mass of a Semi Circle (without integration)
QUESTION [5 upvotes]: Possible Duplicate:
Solving $2x - \sin 2x = \pi/2$ for $0 &lt; x &lt; \pi/2$ 

For fun, I was trying to solve this problem without doing calculus.  After dinking around with it for a while, I came across the following term and don't quite know how to solve it without guessing:
$$a - \sin{(a)} = \frac{\pi}{2}$$
Where $a$ will be the angle of the chord that will allow me to solve for the height - in theory :)

REPLY [4 votes]: You can rewrite the expression as $$\sin(a) =a - \frac{\pi}{2}$$ so that each side of the equality is a function of $a$, each of which can be graphed (on the same graph!). 
The LHS can be graphed as $$f(a) = \sin(a)$$ the graph of which you should know well, 
and the RHS can be graphed as $$g(a) = a - \frac{\pi}{2}$$ Clearly, the graph of $g(a)$ is simply a line, with y-intercept $\left(-\frac {\pi}{2}, 0\right)$ and slope $= 1$.  
You can use your graphs, then, to approximate any potential solutions; that is, find any point(s) of intersection.  From that, you can probably "trouble shoot" with your calculator to make this approximation more precise.
I'll include graphs below: