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 Chief Delphi Geometry Problem
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#1
03-09-2017, 08:56 PM
 Ian Curtis Best Available Data FRC #1778 (Chill Out!) Team Role: Engineer Join Date: Feb 2005 Rookie Year: 2004 Location: Snohomish Posts: 2,554
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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#2
03-09-2017, 09:54 PM
 Cesar Mugnatto Registered User FRC #3932 (The Dirty Mechanics) Team Role: Mentor Join Date: Mar 2017 Rookie Year: 2017 Location: Boca Raton FL Posts: 4
Re: Geometry Problem

θ = inverse sin((b-c) / a)
#3
03-09-2017, 09:57 PM
 Cesar Mugnatto Registered User FRC #3932 (The Dirty Mechanics) Team Role: Mentor Join Date: Mar 2017 Rookie Year: 2017 Location: Boca Raton FL Posts: 4
Re: Geometry Problem

Here is an example of the problem:

#4
03-09-2017, 10:02 PM
 GeeTwo Technical Director AKA: Gus Michel II FRC #3946 (Tiger Robotics) Team Role: Mentor Join Date: Jan 2014 Rookie Year: 2013 Location: Slidell, LA Posts: 4,394
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
• a² - 2ab cotθ + b² cot²θ = c² csc²θ
• a² - 2ab cotθ + b² cot²θ = c² + c² cot²θ
• (a²-c²) - 2ab cotθ + (b² -c²) cot²θ = 0
• cotθ = (2ab ± √(4a²b² - 4 (b² -c²) (a²-c²) ))/2(b² -c²))
• θ = cot⁻¹((ab ± c√(a²+b²-c²))/(b² -c²))
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Last edited by GeeTwo : 03-11-2017 at 06:42 AM. Reason: Added open parenthesis on penultimate line
#5
03-09-2017, 10:09 PM
 Cesar Mugnatto Registered User FRC #3932 (The Dirty Mechanics) Team Role: Mentor Join Date: Mar 2017 Rookie Year: 2017 Location: Boca Raton FL Posts: 4
Re: Geometry Problem

Quote:
 Originally Posted by Cesar Mugnatto θ = inverse sin((b-c) / a)

θ = sin⁻¹((b-c) / a)
#6
03-09-2017, 10:42 PM
 Ian Curtis Best Available Data FRC #1778 (Chill Out!) Team Role: Engineer Join Date: Feb 2005 Rookie Year: 2004 Location: Snohomish Posts: 2,554
Re: Geometry Problem

Quote:
 Originally Posted by Cesar Mugnatto 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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CHILL OUT! | Aero Stability & Control Engineer
Adam Savage's Obsessions (TED Talk) (Part 2)
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#7
03-09-2017, 11:07 PM
 Ether systems engineer (retired) no team Join Date: Nov 2009 Rookie Year: 1969 Location: US Posts: 8,680
Re: Geometry Problem

...
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#8
03-09-2017, 11:17 PM
 Cesar Mugnatto Registered User FRC #3932 (The Dirty Mechanics) Team Role: Mentor Join Date: Mar 2017 Rookie Year: 2017 Location: Boca Raton FL Posts: 4
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.
#9
03-10-2017, 06:49 AM
 GeeTwo Technical Director AKA: Gus Michel II FRC #3946 (Tiger Robotics) Team Role: Mentor Join Date: Jan 2014 Rookie Year: 2013 Location: Slidell, LA Posts: 4,394
Re: Geometry Problem

Quote:
 Originally Posted by Ian Curtis 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
#10
03-10-2017, 10:48 AM
 Ian Curtis Best Available Data FRC #1778 (Chill Out!) Team Role: Engineer Join Date: Feb 2005 Rookie Year: 2004 Location: Snohomish Posts: 2,554
Re: Geometry Problem

Quote:
 Originally Posted by GeeTwo 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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Adam Savage's Obsessions (TED Talk) (Part 2)
It is much easier to call someone else a genius than admit to yourself that you are lazy. - Dave Gingery

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