Systems of Equations/Rearranging Functions Help
Equations in a nice format: https://imgur.com/CAqXC7C
Equations:
h=sqrt((x^2+y^2-2xycos(165))-(P/2)^2)
m=h-L
a=(arctan(m/(9/2)))-20
S=sqrt((m^2)+(P/2)^2)
x=sqrt(y^2+S^2-2Sycos(a)
b=arctan((m/(P/2))-5
y=sqrt(S^2+x^2-2Sxcos(b)
Goal:
y(P,L) = ? & x(P,L,y) = ?
or
y(P,L,x) = ? & x(P,L) = ?
Note: The actual end goal is to get these equations into a format that can be put in to an excel spreadsheet such that values for P and L can be input, and values for x, y, and h will be calculated. So if anyone knows of a different way to do that, that would work too.
1 Answer
If you add the eqs for x² and y² and subtract x²+y² on both sides you get 0 = S (S - x cos b - y cos a).
With S² = m² + P²/4 we can exclude S=0 and therefore S = x cos b + y cos a.
(You can also subtract them to get x²-y² = S ( x cos b - y cos a ) but I see no immediate use in that.)
Let's note that a,b are just a shortcut for something depending on m,P
You also have S² = m² + P²/4, so you also have the equation m² = (x cos b + y cos a)² - P²/4
But I don't see how we can go further, analytically...
Also numerically I did not get any enlightening... Do you have a rough idea/suggestion of what possible values for at least some of these variables could be? (To use as a reasonable starting point for a numerical solution, for example. When I try random values between 0.5 and 2 I get somewhat strange results, although I can find numerical solutions for given L & P.)
Also, I wonder whether your equations are correct, in particular : a = t - 20 and b = t - 5,
which are are used as argument in cos(.), and where t = arctan(2m/P) is an angle :
that should indeed be an angle, but how can appear -20 and -5 here ?
Are you sure that these are values (in radians ?) that must be subtracted from the angle t = arctan(2m/P) ?
PS: now that I think it over, and esp. in view of the cos(165), I wonder whether they are not rather in degrees, which would of course change things quite a bit. But then you should please write 165° (= 165*pi/180), and also a = (...) - 20°, b = (...) - 5°.
M F H
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Sorry, yes the angles are in degrees. As for having a reasonable solution, here is a link to the graph with adjustable sliders for P and L. In this graph, the values I am looking for are the x and y of the upper intersection point. https://www.desmos.com/calculator/sogziflxzm
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What I am hoping for is an equation for the x or y value of that intersection point that is not dependent on the other ie y(P,L) or x(P,L)
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This type of time-consuming problems should come with a bounty, otherwise you may not get a response.
Okay thank you. I'll repost with a bounty as a last resort, if none of the other avenues pan out, then.
I figured it was a long shot, anywa
[I guess the 9/2 in the pasted formula for a is a typo and should be P/2....?]
Are the values that appear in the cos(...) (namely, cos(165) and cos(arctan - 20), cos(arctan - 5) in radians, or are they in degrees?
the angles are in Degrees