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Regarding Jack's comments:
David-
I don't recognizem my solution as identical to yours. Either
one of us has made an error, or I gave up to soon on the algebra.
Jack
and
David, one of us has made an error. The solutions are identical in
the special case of q=0, which explains why we got nearly identical
solutions for the case of small q. We can communicate privately
until we agree on a solution.
Regards,
Jack
Jack, we *already* agree on a solution. Your implicit equation for
the angle a is clearly not *formally* identical to my explicit
formula for it, but your equation *is* equivalent to mine in the
sense that they both have the same value of a as the solution for a
given value of q. In fact, it is fairly straightforward to derive
my formula from your equation. Neither of us made an error. You
just gave up on the algebra a bit to soon.
Below are a few more details of the steps in the derivation that I
outlined in my previous response to Ken Fox (except I am now using
Jack's notation for the variables rather than my/Anthony's notation).
First let's define s == 1 - cos(2*a). Substitution of 1 - s for
cos(2*a) and s/2 for sin^2(a) into Jack's equation 2.) gives (after
dividing both the numerator and denominator on the RHS by sin(a):
1 - s = q/(1 + sqrt(1 + 2*q/s))
Multiplication on both sides by the denominator gives:
(1 - s)*(1 + sqrt(1 + 2*q/s)) = q
Isolating the sqrt() term gives:
(1 - s)*sqrt(1 + 2*q/s) = q - 1 + s
Squaring out both sides and canceling the common terms on both sides
gives:
1 + 2*q/s - 4*q = (q - 1)^2
Solving for the only appearance of s in the above equation gives:
s = 2*q/((q - 1)^2 + 4*q - 1)
Simplifying the denominator and canceling the common factor of q in
the fraction on the right gives:
s = 2/(q + 2)
This means that 1 - s = 1 - 2/(q + 2) = q/(q + 2)
Now since we had defined s as 1 - cos(2*a) this means that we can
write a as:
a = (1/2)*arccos(1 - s) = (1/2)*arccos(q/(q + 2)) QED
Also, if we are willing to write this in terms of 1/q (to Ken Fox's
chagrin about what happens when q --> 0) we can also write:
a = (1/2)*arccos(1/(1 + 2/q))
David Bowman