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Question: Solving a simultaneous equation with angles and distances

Question:Solving a simultaneous equation with angles and distances

miasharmse 13 Maple
August 25 2005
1
After analysing a sketch I came up with two equations describing the sketch.

My first equation looks like: eq1 := tan(X/2) = Vt/Hr;
My second equation looks like: eq2 := tan(X) = (V-Vt)/Hl;

I know the values for Hr, Hl, V so I am left with two unknown values which is angle X and distance Vt.

I have typed these equations in Maple 8 and when I substitute the values Hr, Hl, V then Maple does come up with the correct solution to either angle X or distance Vt. My problem is that I do not want the specific answer to this problem. I need a general answer. That means I want an equation where I can put in any legal values for Hr, Hl, V and then the equation will calculate either Vt or X for me.

With the current sketch I have the following values:

> Hl = 10;
> Hr = 42;
> V = 50;

If I solve eq1 for X I get: X = 2*arctan(Vt/Hr)
If I solve eq2 for X I get: X = arctan((V-Vt)/Hl)

When substituting the above values I get:
X = 2*arctan(1/42*Vt)
X = -arctan(-5+1/10*Vt)

This means that
2*arctan(Vt/Hr) = arctan((V-Vt)/Hl) OR with the values substituted
2*arctan(1/42*Vt) = -arctan(-5+1/10*Vt)

Having the values for Hr, Hl, V given above it must be possible to now solve for Vt, and so it is. The answer that Maple gets is 27.362244 which is absolutely correct. My problem is that I also want to solve for Vt when I do not know the values for Hr, Hl, V. Thus when having the equation "2*arctan(Vt/Hr) = arctan((V-Vt)/Hl)" I want to solve for Vt as well, but now Maple gives me a 20page formula which is of no use to me at all. What I want is a short formula where I can substitute the values Hr, Hl, V and then the formula must give me Vt. If you would need more information I can send the sketch I have made for this problem by e-mail.
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