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Unread 03-09-2017, 08:56 PM
Ian Curtis Ian Curtis is offline
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Geometry Problem

I ran into a very practical problem math building a gate this weekend. Basically I had two 2x4s running parallel a certain distance apart, and I wanted to cut a cross piece that for aesthetic reasons ran from one end to the other. This is obvious a simple SOHCAHTOA for a line, but for a beam of finite width it isn't so simple. I ended up solving it practically (physically do it, and measure), but now I can't make the math work and it is bothering me.

I end up with an equation, but it has two trig functions that I can't reduce with a trig identity, so it isn't directly solvable by an inverse trig function. It seems like it should have an obvious direct solution...
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Unread 03-09-2017, 09:54 PM
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Re: Geometry Problem

θ = inverse sin((b-c) / a)
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Unread 03-09-2017, 09:57 PM
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Re: Geometry Problem

Here is an example of the problem:

https://www.geogebra.org/o/Jh8DADfg
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Unread 03-09-2017, 10:02 PM
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Re: Geometry Problem

Not the prettiest way, but a single trig identity and the quadratic equation will get you there:
  • a = b cotθ + c cscθ
  • a - b cotθ = c cscθ
  • a - 2ab cotθ + b cotθ = c cscθ
  • a - 2ab cotθ + b cotθ = c + c cotθ
  • (a-c) - 2ab cotθ + (b -c) cotθ = 0
  • cotθ = (2ab √(4ab - 4 (b -c) (a-c) ))/2(b -c))
  • θ = cot⁻((ab c√(a+b-c))/(b -c))
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Last edited by GeeTwo : 03-11-2017 at 06:42 AM. Reason: Added open parenthesis on penultimate line
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Unread 03-09-2017, 10:09 PM
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Re: Geometry Problem

Quote:
Originally Posted by Cesar Mugnatto View Post
θ = inverse sin((b-c) / a)
Using symbols instead of text...

θ = sin⁻((b-c) / a)
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Unread 03-09-2017, 10:42 PM
Ian Curtis Ian Curtis is offline
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Re: Geometry Problem

Quote:
Originally Posted by Cesar Mugnatto View Post
Here is an example of the problem:

https://www.geogebra.org/o/Jh8DADfg
This tool is super cool, but your answer doesn't appear to work? sin^-1 ((6-1.2)/10) is 28.6 degrees, not 36.87 degrees? One of the solutions to GeeTwo's quadratic appears to be the correct answer.

I probably knew this once upon a time, what does the other solution to that quadratic represent?
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Unread 03-09-2017, 11:07 PM
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Re: Geometry Problem

...
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Unread 03-09-2017, 11:17 PM
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Re: Geometry Problem

Ether's solution is correct. Mine started out the same way but in the process I dropped / forgot to cross multiply the right side by cos theta.
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Unread 03-10-2017, 06:49 AM
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Re: Geometry Problem

Quote:
Originally Posted by Ian Curtis View Post
This tool is super cool, but your answer doesn't appear to work? sin^-1 ((6-1.2)/10) is 28.6 degrees, not 36.87 degrees? One of the solutions to GeeTwo's quadratic appears to be the correct answer.

I probably knew this once upon a time, what does the other solution to that quadratic represent?
If you substitute c with -c in either solution, it becomes the other. So the other solution is the answer if you did the problem with c being outside of a, or equivalently if the inside edges of the board (rather than the outside edges) were a apart at span b.
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Last edited by GeeTwo : 03-10-2017 at 06:50 AM. Reason: swapped a and b originally
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Unread 03-10-2017, 10:48 AM
Ian Curtis Ian Curtis is offline
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Re: Geometry Problem

Quote:
Originally Posted by GeeTwo View Post
If you substitute c with -c in either solution, it becomes the other. So the other solution is the answer if you did the problem with c being outside of a, or equivalently if the inside edges of the board (rather than the outside edges) were a apart at span b.
Ah, yes. Thanks! I could get to the top of your post, but I did not remember the trick (square it!) to get to a single trig function you could invert.

I'm still a little surprised the solution is so complicated, I was sure I was missing something obvious and it was going to end up with something like Cesar's original post.
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