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Wikipedia:Reference desk/Archives/Mathematics/2008 June 4

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June 4

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Laws of Cosines

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I understand how the laws of cosines work, but what I can't figure out is how, when they give you all three sides of a triangle, you do the problem. I get to a certain point and then get stuck... Could someone go step by step through an example and explain what they are doing each step? --Devol4 (talk) 06:45, 4 June 2008 (UTC)[reply]

Easy!

side_A = 10 cm
side_B = 7 cm
angle_C = 20 degrees

Find the length of side_C

Solution:

side_C ^ 2 = side_A ^2 + side_B ^2 - 2 * side_A * side_B * Cosine(angle_C)
side_C ^ 2 = 10^2 + 7^2 - 2 * 10 * 7 * Cosine(20 degrees)
side_C ^ 2 = 100 + 49 - 140 * Cosine(20 degrees)
side_C ^ 2 = 100 + 49 - 140 * 0.9397
side_C ^ 2 = 100 + 49 - 131.558
side_C ^ 2 = 17.442
thus after we take the (positive) square root
side_C = 4.176 cm

The end. 122.107.152.72 (talk) 08:42, 4 June 2008 (UTC)[reply]

I think he actually wanted to know how it works when you have all three sides known. Suppose you know the length of sides a and b, which are adjacent to angle C, and you also know side c which is opposite C. Then you just rearrange the formula to put cosC on its own. Then all the values on the opposite side you should know, and so you have the exact value of cosC. Take the inverse cosine of each side and you're there. -mattbuck (Talk) 08:59, 4 June 2008 (UTC)[reply]

Alternatively, you can be given all three sides and are asked to find an angle.

side_A = 10 cm
side_B = 7 cm
side_C = 4.176 cm

Find the value of angle_C

Solution:

side_C ^ 2 = side_A ^2 + side_B ^2 - 2 * side_A * side_B * Cosine(angle_C)

After moving "side_A ^2 + side_B ^2" across to the Right Hand Side

side_C ^ 2 - side_A ^2 - side_B ^2 = - 2 * side_A * side_B * Cosine(angle_C)
Next we multiple both sides with negative one (-1)
side_A ^2 + side_B ^2 - side_C ^ 2 = 2 * side_A * side_B * Cosine(angle_C)
Next we divide both sides with "2 * side_A * side_B"
(side_A ^2 + side_B ^2 - side_C ^ 2) / (2 * side_A * side_B) = Cosine(angle_C)

Now we put the numerical values in

(100 + 49 - 17.442) / ( 2 * 10 * 7) = Cosine(angle_C)
131.558 / 140 = Cosine(angle_C)
0.9397 = Cosine(angle_C)
Now I look up my ArcCosine Table for the entry 0.9397 to get
19.99 degrees = angle_C

Thank you Ohanian (talk) 08:57, 4 June 2008 (UTC)[reply]

Alright, thanks. I managed to figure out that I was typing it into the calculator wrong... Bad me. But thanks for the help, I now understand it better and it should help me in the test next period.

--Devol4 (talk) 11:43, 4 June 2008 (UTC)[reply]

ax^n+bx+c

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I think this is a special class of equation that has been studied a bit, but can’t figure out what they're called. Equations of the form ax^n+bx+c. —Preceding unsigned comment added by 130.127.186.122 (talk) 19:22, 4 June 2008 (UTC)[reply]

Those would most likely be polynomials, assuming that n is a non-negative integer. — Lomn 19:32, 4 June 2008 (UTC)[reply]
Presumably the correspondent was looking for a more specific word for a polynomial with no term between the largest power of x and the linear term though?--130.88.123.142 (talk) 14:01, 9 June 2008 (UTC)[reply]

A Mathematical Puzzle

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You have 25 horses, and a track on which you can race five of them at a time. You can determine in what order the horses in a race finished, but not how long they took, and so can not compare times from one race to another. A given horse runs at the same speed under all circumstances, and no two horses run at the same speed. How many races does it take to find the three fastest? —Preceding unsigned comment added by 171.69.159.185 (talk) 22:07, 4 June 2008 (UTC)[reply]

It was never completely answered, though: no-one proved that you can't do it in six races. Algebraist 22:26, 4 June 2008 (UTC)[reply]
I still say I could do it in only 53,130 races. -mattbuck (Talk) 22:32, 4 June 2008 (UTC)[reply]
Well done. Algebraist 22:37, 4 June 2008 (UTC)[reply]
I have no idea. But if you think bout it like this: 1st race:5 horses, who wins means nothing because they could all be the fastest or all be the slowest, so take the top three against 2,so far, you have 7 horses and you know your relative fastest three. continue the top three against next two and you get 11 races. But I probably did something wrong, and knowing how everything in math has to do with patterns I'm going to say 15, 1+2+3+4+5, or the 5th triangle number.--Xtothe3rd (talk) 03:12, 5 June 2008 (UTC)[reply]
As mentioned in the archived discussion, it's fairly easy to see that seven races are enough for the top three, and that five can't even give you the top one. Probably six aren't enough, but I at least can't prove it. The problem is that six races are enough to demonstrate which are the top three if you somehow know already, so the naive proof fails. Algebraist 07:36, 5 June 2008 (UTC)[reply]
If you decided to follow the following strategy - which in general is *not* optimal;
  • Race the first 5 horses
  • Race the next four horses with the current overall 3rd placed horse, based on the races already done.
  • If the 3rd placed overall horse wins, race it again against the next four horses.
  • If the 3rd placed overall horse does not win, race the current overall fastest and 2nd fastest against the first three that beats the 3rd overall fastest.
If it turns out that the winner, runner up, and 3rd place of race one are actually the fastest three horses, then, if you followed the above strategy, then the top three horses would be known after only 6 races. So for some cases, it would be possible to do it in less than 7. Richard B (talk) 14:06, 5 June 2008 (UTC)[reply]
You could look at it as a combinatorial game: one player chooses which horses to race in each round and tries to deduce the top three, the other chooses the outcomes of the races (subject to consistency with earlier rounds) and tries to stop the first player from succeeding in six rounds or less. It's obvious that one player or the other must have a winning strategy, you just need to find out which one it is. Anyone have a good game tree solver? —Ilmari Karonen (talk) 14:46, 5 June 2008 (UTC)[reply]
By the way, just to clarify, does the task require determining the order of the fastest three horses, or just which ones they are? That is, would a method that allowed one to determine that A, B and C are the fastest three horses, but not whether A is faster than B, qualify? —Ilmari Karonen (talk) 14:46, 5 June 2008 (UTC)[reply]
Having thought about this while on the bus, I can now prove that six races are not enough, at least not if the fastest three horses have to be ranked. In fact, I can prove a stronger result: six races are not enough to uniquely determine both the fastest and the second-fastest horse. The proof proceeds as follows:
To determine the fastest horse, 24 contenders for the first place must be eliminated. This can be done in six races, but only just: each race can eliminate at most four contenders, for a total of 6 × 4 = 24. To achieve this, every horse in each race must be a contender for the first place; that is, none of them may have lost an earlier race.
Thus, the sixth and final race must be between five horses that have not lost any previous race. Further, at least one of them must've won an earlier race; after five races there can be at most four horses who haven't yet participated in any race. Call that horse X. If horse X wins the final race, there will be at least two other horses that have only ever lost to X: the one that came second in the last race, and the one that came second in the previous race that X won. Thus, the second-fastest horse will not be uniquely determined in only six races, Q.E.D. —Ilmari Karonen (talk) 20:37, 5 June 2008 (UTC)[reply]
Looks good to me. --Tango (talk) 21:11, 5 June 2008 (UTC)[reply]
How about if they don't have to be ranked? Black Carrot (talk) 01:13, 6 June 2008 (UTC)[reply]
This was the interpretation I took, that it asks only for the 3 fastest though not necessarily ordering those 3. Does this change the question significantly? 98.221.167.113 (talk) 02:08, 6 June 2008 (UTC) whoops, mine Someletters<Talk> 02:09, 6 June 2008 (UTC)[reply]
Probably not, but it invalidates the proof technique I used (consider all strategies for finding the winner in six races, then show that none of them is able to always determine the second place too). By the way, note that my proof above has a minor, fortunately inconsequential omission: to eliminate four contenders for the first place in one race, it's not necessary for all of the entrants to have never lost before, merely for all but the winner. However, since one cannot determine the winner of such a race in advance, there's no way to take advantage of this. —Ilmari Karonen (talk) 11:42, 6 June 2008 (UTC)[reply]
But what about my situation above? I chose a different strategy in deciding who races - in picking a horse for a subsequent race that *had* already been beaten (into 3rd place) - and if you were fortuitous enough that the fastest three horses were all selected to take part in the first race - then the top three horses can be both determined - and ranked - in 6 races.Richard B (talk) 12:15, 6 June 2008 (UTC)[reply]
That works only if you already know which the fastest three horses will be, or if you're really lucky. The original request was for a method that would allow determining the fastest three horses in six races in every case, no matter how unlucky you might be. —Ilmari Karonen (talk) 13:01, 6 June 2008 (UTC)[reply]
(ec) I think a slight modification will work, though. This is off the top of my head, and so probably more convoluted than necessary:
Even if you don't need to order the fastest three horses, that still means you need to eliminate at least 22 contenders for the first place (since, clearly, any horse who might be the fastest could also be one of the three fastest). In five races you can eliminate at most 20, which means that, before the last race, there will be at least five horses who could be fastest (not to mention several more who could be second- or third-fastest) and again, at least one of them (call it X) must've won at least one earlier race.
If there are more than five, the last race can disqualify at most two of them from being among the fastest three, which is obviously not enough. If there are exactly five, that means exactly 20 must've been eliminated in the first five rounds, and thus no horse can have lost more than one race. Thus, the horse who came second in the previous race that X won has not lost to any other horse, and is thus still a contender for the second place. We'll call that horse Y.
If at most three of the five remaining first-place contenders participate in the last race, we'll assume they score among the fastest three in that race, and will thus all remain contenders for the first three places overall. If four do, we'll assume three of them take the first three places in that race, and so there will still be at least four contenders for the first three places overall. Finally, if all of the five first-place contenders participate in the last race, then we'll assume that the winner of that race will be X. Then Y, together with the first three horses from the last race, will still be a contender for the first three places overall. In each case, there will be at least four horses left who might be among the fastest three, Q.E.D. —Ilmari Karonen (talk) 12:59, 6 June 2008 (UTC)[reply]
Yup. Sounds good. Black Carrot (talk) 18:38, 6 June 2008 (UTC)[reply]
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