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Solving $2x - \sin 2x = \pi/2$ for $0 < x < \pi/2$

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 :)

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I should plug this in as-is and see if either the sin or the angle disappears. I could see this turning in 4R/3Pi if I worked it through. –  Michael Dorgan Jul 1 '11 at 17:22
This is essentially the same equation as in the question here math.stackexchange.com/q/48865/2906 –  Mark Bennet Jul 1 '11 at 17:22
Not quite, Sin(2a) implies a double angle theorm with more ways to changes the terms around. It's actually really close to the Sin(x) = 1 - x from (26538), which answers stated it was transdendal and guessing/iterating was the way to solve it. But, because the value we're playing with is a Pi multiple, I thought there might be a shortcut or two I wasn't aware of. –  Michael Dorgan Jul 1 '11 at 17:40
This is very closely related to solving $\cos x = x$ (see the duplicate thread in earlier comment) and also see this: Dottie Number –  Aryabhata Jul 1 '11 at 17:49
@Michael look carefully and you will see that there are factors of two on both sides of the equation I linked so the are essentially the same. There is an interesting point that the two formulae arise in different ways, and suggest different methods of approach. –  Mark Bennet Jul 1 '11 at 17:54
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marked as duplicate by Aryabhata, Willie Wong Jul 1 '11 at 18:51

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

1 Answer

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:

enter image description here

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enter image description here

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Was hoping for an algebraic solution, but this will do as a fall back. Thanks! –  Michael Dorgan Jul 1 '11 at 17:47
Sometimes just having an estimated solution helps you then construct an algorithm, if it is possible to do so, since you then have your "starting point", and an approximate "ending point" to work with. Also, simply having a visual representation of what's "going on" with a given expression is helpful. –  amWhy Jul 1 '11 at 17:54
Agreed. I'll give you your cookie then :) –  Michael Dorgan Jul 1 '11 at 17:57
The solution is given by $a = \frac{\pi}{2} - D$ where $D$ is the Dottie Number. See my comments to this question. If you agree that this is a dupe, please cast your vote. This is the third such question... (see Mark's comment for the second one). –  Aryabhata Jul 1 '11 at 18:29
Guys, the algrebra is wrong here, it should be a - π/2, the signs are backwards in this answer. –  Michael Dorgan Jul 1 '11 at 18:37
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