Equivalent equations and not equivalent answers?

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Discussion Overview

The discussion revolves around the discrepancies in the area calculations of a SSA triangle using two different equations. Participants explore the reasons why these equations, which should yield equivalent results, produce different answers when applied to the same triangle parameters.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents a scenario with a SSA triangle and calculates the area using two different formulas, noting a discrepancy in the results.
  • Another participant suggests that using more accurate values for angles might resolve the differences in area calculations.
  • A participant mentions that rounding to two digits might limit the accuracy of the results, implying that the precision of input values affects the output.
  • There is a discussion about the nature of calculator precision and how it may differ from mathematical representations, with one participant asserting that the calculator's operations can yield slightly different results than expected.
  • A later reply indicates that using a high-precision tool like Mathematica produced results that were very close, suggesting that the discrepancies may be due to rounding or precision issues rather than fundamental differences in the equations.

Areas of Agreement / Disagreement

Participants express differing views on the reasons for the discrepancies in the area calculations, with some attributing it to rounding and others questioning the equivalence of the equations under certain conditions. The discussion remains unresolved regarding the exact cause of the differing results.

Contextual Notes

Participants note that the accuracy of angle measurements and rounding practices may significantly influence the results, but there is no consensus on how these factors interact with the equations used.

mewmew
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Ok, What do you do when two equations that should yield the same answer dont? For example when finding the area of a SSA triangle:
A= 25 degrees
side a = 100
side b = 200

Now, I know this triangle gives two possible answers but I am just concentrating on answer 1, when using the law of sines you get
B= 57.6973 degrees
which leaves C= 97.3027 degrees

Now, if you use the equation:

(a^2*sinB*sinC) / (2sinA) You get 9,958.48 inches^2

But, if you use .5(a)(b*sinC) which should work also, you get 9,925.88 inches^2

If these equations are equivalent then why do they give different answers? Is it just calculator rounding?
 
Last edited:
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Have you tried using more accurate numbers? (e.g. B isn't exactly 58 degrees)
 
Sorry, I just rounded a few numbers to make it easier to read, but using my ti-89 and using the numbers it gives me I still get equation
1. = 9958.48 inches^2 and equation
2. = 9918.88 inches^2
 
Have you tried using more accurate numbers? (e.g. B isn't exactly 58 degrees)
 
I don't know that it is "calculator" rounding. Surely your calculator gives better than 2 digit accuracy: but it is reasonable for you to round to 2 digits: you were given A as 25 degrees: 2 digits accuracy. Of course, after you have done that, there is no point in expecting more than 2 digits accuracy in your answer: rounded to 2 digits, you get 9900 or, better, 9.9x103.
 
Have you tried using more accurate numbers? (e.g. B isn't exactly 58 degrees)
The answers I gave in my second post are from using the exact numbers my calculator gave me to solve it. I just don't understand why two equations that should give the same answer dont? I sort of understand why 2 digit accuracy would follow through to my answer I guess, it just seems counterintuitive that two equivalent equations give different answers.
 
The answers I gave in my second post are from using the exact numbers my calculator gave me to solve it.

No they're not. You used 58 for B and 97 for C.



it just seems counterintuitive that two equivalent equations give different answers.

It's not so bad once you get used to the idea. The + sign on your calculator really is something slightly different than the + sign you write in mathematical formulae.
 
Hurkyl said:
No they're not. You used 58 for B and 97 for C.

It's not so bad once you get used to the idea. The + sign on your calculator really is something slightly different than the + sign you write in mathematical formulae.

Sorry, I edited my first post to show that I used a more accurate answer than just 58 and 97 when getting my answers, I should have just done that in the first place but used rounded answers so my post would look nicer. But anyways, I think you guys have explained it well enough for me. It was just something I had noticed more than once in more than one kind of formula that always bothered me. Thanks
 
Well, I hate to keep this thread going but I did a little test with Mathematica. I took all of my values to a precision of 100,000 decimal places and then computed the two answers. The answers I got of course where more precise than my calculator and gave me very similar results, they are the same down to somewhere around 25 or so decimal places. I just thought you guys would like to know, and thanks for the help.
 

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