Equivalent equations and not equivalent answers?

[mewmew]

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?

 

[Hurkyl]

Have you tried using more accurate numbers? (e.g. B isn't exactly 58 degrees)

 

[mewmew]

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

 

[Hurkyl]

Have you tried using more accurate numbers? (e.g. B isn't exactly 58 degrees)

 

[HallsofIvy]

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.

 

[mewmew]

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.

 

[Hurkyl]

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.

 

[mewmew]

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

 

[mewmew]

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.