Recently I had occasion to work out a problem involving right triangles (long story short, I was doing a paper folding project and had to determine how far down the side of a sheet of paper to make a fold). Since I had the length of two sides of a right triangle I could determine the third. I originally did this using the Pythagorean theorem; later I decided to confirm it with trigonometry and got the same result. It occurs to me that the same input yielding the same result means that the two methods have to somehow be logically equivalent. How would you demonstrate this?
Ultimately, you’re making use of the Pythagorean trig identity, sin[sup]2/sup + cos[sup]2/sup = 1.
Of course, this identity is derived from the Pythagorean Theorem in the first place, so it’s not really anything new.
The Law of Cosines defaults into Pythagorean theorum for a right triangle … that’s why it’s easy to remember:
a[sup]2[/sup] + b[sup]2[/sup] - 2ab cos C = c[sup]2[/sup] => a[sup]2[/sup] + b[sup]2[/sup] = c[sup]2[/sup] when C = 90º
It is reassuring to note that in mathematics, sanity prevails. (Usually.)
I’ve encountered situations several times where a problem obviously had a unique well-defined solution, but there were two distinct ways to analyze the problem, leading to two distinct ways to solve it. If sanity prevails, the several solutions ought to lead to the same final answer.
So it’s reassuring when that actually happens.
Case in point: Typical simple permutations/combinations type of problem:
A largish group (of say, 30 people) wishes to choose a committee of six to do something: The committee is to have a chairperson, a secretary, and four other at-large members. How many ways can such a committee be chosen from the group? (To clarify: It’s not just how many groups of six can be chosen from the group of 30. Every way a distinct person can be chairperson or secretary counts as a separate “way”.)
I immediately saw two distinct strategies by which such a committee could be chosen:
(A) Choose a chairperson from the group of 30, then a secretary from the remaining 29, then four more from the remaining 28.
(B) Choose a group of six from the group of 30, then within that group of six, choose one to be the chairperson and another to be the secretary.
I’ll leave it as an exercise for the reader to work out the numbers. But note this: The mathematics to describe method (A) is quite different from the mathematics to describe method (B). Yet it’s apparent that there can only be one solution to the problem, so methods (A) and (B) ought to end up with the same answer if mathematics is trustworthy.
I worked it out both ways, and sure enough, got the same answer. It should certainly be possible to work out an algebraic demonstration that the two methods are really just the same, perhaps with some of the operations happening in a different order or something.
I’ll make another post with another case I saw once where something like this happened. – It may take me a few minutes to remember it in enough detail . . .
Second case in point:
Two events are said to be independent, loosely speaking, if the occurrence or non-occurrence of either event has no effect on the probability of the other event happening.
The usual formal mathematical description of this is:
P(A | B) = P(A)
Taking this literally, this says: The probability of event A given that event B has occurred is equal to the probability that event A occurs (with no mention of B occurring or not).
But the way I always thought of it was a bit different: The probability of event A given that B has happened is the same as the probability of event A given that B has not happened. Translating this literally into an equation gives:
P(A | B) = P(A | B’)
These two formulas say two different things, and it’s not immediately obvious that they would be equivalent. Yet they express two different notions of what “independent events” means that seem like they might be equivalent.
I mentioned this to my statistics professor. He got interested. Grabbing a pencil and paper, he tried to develop an algebraic proof that the two formulas are equivalent. After just two or three steps, he got stuck. Simultaneously, I grabbed a pencil and paper and tried doing the same – but I didn’t get stuck where he did. With a little bit of clever factoring, I was able to complete the proof.
Again, two views on what independence means that are (hopefully but not too obviously) equivalent, leading to two different formulas defining independence – and sure enough, with a little algebra it was proved that they really are the same.
The Pythagorean Theorem is so well-known as to be cliché, but still underlies much of importance in applied math.
Here is an extreme, exotic example of distinct derivations.
There are exactly 10 sets of natural numbers, all differing by at least 2, which sum to 13.
13 = 13
13 = 12+1
13 = 11+2
13 = 10+3
13 = 9+4
13 = 9+3+1
13 = 8+5
13 = 8+4+1
13 = 7+5+1
13 = 7+4+2
And similarly, there are exactly 10 bundles of natural numbers, all of the form 5k ± 1, which sum to 13.
13 = 11+1+1
13 = 9+4
13 = 9+1+1+1+1
13 = 6+6+1
13 = 6+4+1+1+1
13 = 6+1+1+1+1+1+1+1
13 = 4+4+4+1
13 = 4+4+1+1+1+1+1
13 = 4+1+1+1+1+1+1+1+1+1
13 = 1+1+1+1+1+1+1+1+1+1+1+1+1
It’s well-known (though only by those who know such things well) that these partition cases are always equal in number, not just for this N=13 example. But proving this with a combinatorial argument is not easy. (I was a bystander to history as my girl-friend was among the first informed when a bijection was discovered. She, a non-mathematician, seemed so excited I thought Goldbach’s conjecture or something had been proved!)
Ah, so trig is a generalization of the Pythagorean Theorem, with right triangles being a special case. Got it.
There are virtually two or more ways of looking at something and it is a reality check to see that they agree. I have just proved something (too technical to go into here) that was easy from one point of view and quite surprising from my original way of looking at it and I am trying to see why it is true from the latter point of view (as it must be).
I think you might find the link in the fact that sin(x) has a known, fixed value determined by the Pythagorean Theorum, more or less.
In other words, sin(27)=.4539904997395 because the Theorum says so. sin(27) doesn’t equal .569467894789 because the Theorum says it ain’t so.
There’s a reason the trig functions are transcendental.
See Post #2.
That is not what what it says. It does say that the sum of the squares of the lengths of the short sides equals the square of the length of the long side, so that as a result if you know any two of those lengths you can calculate the third. (Note, you do not need to calculate the angle to do this.)
This is true, but I don’t see how it immediately follows from the Pythagorean Theorem.
Well, I was certainly excited myself when I proved Goldbach’s conjecture a couple years ago. The proof was quite simple and straightforward, too, although just a bit too long for this text box to contai
(That’s also why no mathematician will ever become famous on Twitter.)
Since there is a fixed relationship regarding the lengths of the sides of right triangles, there will be a fixed relationship involved in comparing two of those lengths as a function of one of the non-right angles. Since any non-right triangle can be turned into two right triangles, this allows us to fix the relationships among sides of non-right triangles as a function of one or more of the angles.
The trigonometric values of sin and cos (and tan and cot and sec and csc!) do not derive from the Pythagorean Theorem. Rather, the Theorem and the values both can be derived from the characteristics of right triangles. That’s why there are so many, many, many different ways to prove the Theorem. For my money, one proof that follows most naturally from the work done on triangles up to the point it is introduced relies upon the relationships of similar triangles and the provable similarity of the two triangles formed by dropping an altitude in a right triangle to the hypotenuse to each other, and to the original triangle. And all of THAT has at some basic level the assumption you’ve made for dealing with Euclidean geometry about parallel lines.
NM
My favorite “counter example” to this, that I would use to confuse my calculus students. Notationally I’m going to use “Int” to indicate the integral sign.
Solve: Int(sin(x)cos(x)dx)
Method 1: let u = sin(x) du=cos(x)dx; Int(sin(x)cos(x)dx)= Int(u du)=u^2= 1/2 sin(x)^2
Method 2: let u = cos(x) du = -sin(x)dx; Int(sin(x)cos(x)dx)= Int(-u du)=u^2= -1/2 cos(x)^2
Method 3: sin(x)cos(x)= 1/2 sin(2x); Int(sin(x)cos(x)dx)= Int(1/2 sin(2x)dx) = -1/4 cos(2x)
So we get three different answers to the same problem?!?!
Where did I make a mistake?
[SPOILER] I didn’t include the +C in my answer
1/2 sin(x)^2 +C = 1/2(1- cos(x)^2) +C = -1/2(cos(x)^2) + (C-1/2)
-1/4 cos(2x) +C = -1/4(cos(x)^2-sin(x)^2) +C = -1/4(1-sin(x)^2–sin(x)^2)+C=-1/2(sin(x)^2)+(C-1/4)
So they all equal 1/2 sin(x)^2 +C just with different values of C.
[/SPOILER]Yes! Another one along those lines (not involving trigonometry):
Int(1/(5x)dx)
Method 1: u = 5x. du = 5dx, so dx = 1/5du. Then Int((1/5)(1/u)du) = 1/5 ln|u| = 1/5 ln|5x|.
Method 2: 1/(5x) = (1/5)(1/x). So Int(1/(5x)dx) = Int((1/5)(1/x)dx) = 1/5 ln|x|.
Why did we get two different answers? Same reason as in Buck Godot’s example.
For anyone who’d like to read up more about these, these are known as “Rogers–Ramanujan identities”.
As to the OP, it depends on what you mean by “logically equivalent”; after all, the very fact that both approaches answer the same question is the logical argument that both are equivalent in terms of yielding the same result. Whatever logical reasoning told you one method would answer the question, combined with whatever logical reasoning told you the other method would answer the question, combine to give you logical reasoning telling you both methods have to yield the same result.
That having been said, as concerns the Pythagorean Theorem and trigonometry, they’re so closely related that whatever reasoning that leads you to think the one or the other is applicable in any particular situation is probably very similar, and indeed essentially identical. I do not even distinguish the Pythagorean Theorem from trigonometry (nor distinguish trigonometry from complex number arithmetic, for that matter). The Pythagorean Theorem is part of trigonometry. Answering your question in more detail would require knowing exactly what use you made of the Pythagorean Theorem, and exactly what alternative “trigonometry” you used to reach the same result.
I don’t see what Post #2 has to do with mine. The P.T. says, in essence, “If two sides are this, then the third side must be that.” The algebra is beside the point, for the purposes of this explanation.
I never suggested one must calculate the angle, which I interpret to mean you have to determine a number to label it with. What I am suggesting is that if the sides measure this and that, and it’s a right triangle, then everything else must be of certain measurements. They’re fixed. They can’t be different. The triangle is sufficiently defined, and the other measurements are forced.
I’m taking a chance that having the OP realize this fact- that once two sides are decided, everything is decided- will give him the “aha” he’s asking for.
It doesn’t immediately follow, mathematically or algebraically. They are related, however, in the sense that the existence of the P.T. means that two sides force a third. Sine, cosine, and tangent are consequences of those facts.
Image in a world, for the sake of argument, in which two sides in a right triangle didn’t conclusively determine the length of the third side. Imagine for a moment if you needed more than that. In such a world, the trig functions wouldn’t exist, as there’s be no fixed relationship between sides, given just an angle. Sin(27) would have multiple answers.
Consider that in “Sin(27)”, the 27 serves no mathematical purpose other than to define which right triangle we’re talking about. It’s no more meaningful than #A52397ZY0 in the Sears catalog makes something a microwave oven. It’s just a label for a triangle with certain attributes. Sine is determined by side lengths. Side lengths are determined by the Theorum. That’s the relationship.
DSYoungEsq said better anyhow:
What he said.
Sometimes in these threads, posters will spend more time trying to convince the OP of something he already believes to be true, instead of trying to help break through the mental block to understanding. I’m not making a mathematical argument; I’m attempting to appeal to the OP’s intuitive senses to aid in comprehension.