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Newsgroups: comp.soft-sys.matlab
Subject: Re: Combination of variables   with a constraint
Date: Wed, 4 Apr 2012 19:27:12 +0000 (UTC)
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"Leyo Joseph" wrote in message <jli2mb\$5b6\$1@newscl01ah.mathworks.com>...
> Hello,
>
> In the following equation, "a" and "alpha" are variables.
>  sqrt((a*sin(aplha))^2+(1-a*cos(alpha)^2)=0.25
>  Many combinations of "a" and "aplha" are solutions to the above equations.
> For example a=0.9835 and aplha -0.2531 is a solution.
> How to find all of the combinations of "a" and "alpha"  that are solution to the above equations.
> Thanks,
- - - - - - - -
I think you mean

sqrt((a*sin(alpha))^2+(1-a*cos(alpha))^2)=0.25

Your expression is missing one right parenthesis.

As you surely are aware, there are infinitely many combinations of 'a' and 'alpha' that will satisfy that equation.  However, it might help to give explicit solutions of each of these variables in terms of the other.  That way you can see the necessary restrictions that are placed on each variable.

cos(alpha) = a/2+(15/32)/a

a = cos(alpha) +or- sqrt(cos(alpha)^2-15/16)

As you can see, alpha will necessarily be restricted by the requirement that cos(alpha)^2 >= 15/16 for real solutions.  Also 'a' will have to lie between 3/4 and 5/4 or between -5/4 and -3/4.

Nevertheless that leaves infinitely many solutions possible.  Just do an ordinary plot of a/2+15/(32*a) against a to see where the curve falls between -1 and +1, remembering that infinitely many alpha's will give the same cosine value and you will get a feeling for the possible "combinations".

Roger Stafford
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