Doesn't the law of cosines prove SAS?

[MegaMoh]

[MATH]c^2 = a^2 + b^2 -2ab\cos(C)[/MATH] for all triangles shows clearly how if [MATH]a[/MATH] is known, [MATH]b[/MATH] is known, and the angle between them [MATH]C[/MATH] is known, you can find the remaining side and rewriting it again for the other sides([MATH]a^2=b^2+c^2-2bc \cos(A)[/MATH]) and solving for the other angle. Why is SAS still considered an axiom by many? I don't think the steps up to the proof of the law of cosines require SAS to be used.

 

[pka]

[MATH]c^2 = a^2 + b^2 -2ab\cos(C)[/MATH] for all triangles shows clearly how if [MATH]a[/MATH] is known, [MATH]b[/MATH] is known, and the angle between them [MATH]C[/MATH] is known, you can find the remaining side and rewriting it again for the other sides([MATH]a^2=b^2+c^2-2bc \cos(A)[/MATH]) and solving for the other angle. Why is SAS still considered an axiom by many? I don't think the steps up to the proof of the law of cosines require SAS to be used.

How philosophical do you what to get? Geometry, some call it axiomatic geometry, is synthetic. That means that it is driven by definitions and axioms alone. Moreover, the axioms are non-numerical. Now that does not mean that numbers are not assigned a measures of things such as angle size or length of line segments; but it does that measure is not intrinsic to the axioms. Trigonometry is numerical by its very nature: a collection of numerical relations in triangles.

 

[MegaMoh]

How philosophical do you what to get? Geometry, some call it axiomatic geometry, is synthetic. That means that it is driven by definitions and axioms alone. Moreover, the axioms are non-numerical. Now that does not mean that numbers are not assigned a measures of things such as angle size or length of line segments; but it does that measure is not intrinsic to the axioms. Trigonometry is numerical by its very nature: a collection of numerical relations in triangles.

from the axioms up until the law of cosine, as far as I know, you don't need SAS at all. Other than that, you don't use "numericals" in proving it either. It doesn't need complications to show that if [MATH]a[/MATH], [MATH]b[/MATH], and [MATH]C[/MATH] are known, everything else is from the law of cosines.

 

[JeffM]

[MATH]c^2 = a^2 + b^2 -2ab\cos(C)[/MATH] for all triangles shows clearly how if [MATH]a[/MATH] is known, [MATH]b[/MATH] is known, and the angle between them [MATH]C[/MATH] is known, you can find the remaining side and rewriting it again for the other sides([MATH]a^2=b^2+c^2-2bc \cos(A)[/MATH]) and solving for the other angle. Why is SAS still considered an axiom by many? I don't think the steps up to the proof of the law of cosines require SAS to be used.

And how do you prove the law of cosines? Remember that you cannot use SAS or any other proposition dependent on it.
The second point in that sentence is important. You must trace back every proposition that you use to prove SAS to seeee whether any theorem used in the SAS derivation ultimately uses anything dependent on SAS.

 

[Dr.Peterson]

You can start geometry from various different sets of axioms. Some take SAS as an axiom; others don't. You may be interested in this post on my blog, putting together several answers to questions about variations between sets of axioms/postulates: Who Moved My Postulate?

As for your specific question, you'd have to prove the Law of Cosines from some specific set of geometrical axioms to see where it comes from. It would be interesting to see such a proof, which would include proving that the cosine function is well-defined, and so on. You can't just talk about "as far as I know".

Actually, now that I think about it, Euclid proved what is essentially the Law of Cosines: https://mathcs.clarku.edu/~djoyce/java/elements/bookII/propII12.html . I'd have to look closely to see whether this serves as an independent proof of SAS, and whether it is fully justified by his axioms.

 

[JeffM]

You can start geometry from various different sets of axioms. Some take SAS as an axiom; others don't. You may be interested in this post on my blog, putting together several answers to questions about variations between sets of axioms/postulates: Who Moved My Postulate?

As for your specific question, you'd have to prove the Law of Cosines from some specific set of geometrical axioms to see where it comes from. It would be interesting to see such a proof, which would include proving that the cosine function is well-defined, and so on. You can't just talk about "as far as I know".

Actually, now that I think about it, Euclid proved what is essentially the Law of Cosines: https://mathcs.clarku.edu/~djoyce/java/elements/bookII/propII12.html . I'd have to look closely to see whether this serves as an independent proof of SAS, and whether it is fully justified by his axioms.

Actually Dr. P, Euclid proves this theorem by using the Pythagorean Theorem and proves the Pythagorean Theorem using SAS, so we cannot go backward to prove SAS from Euclid's version of the law of cosines. Vicious circle.

 

[MegaMoh]

You can start geometry from various different sets of axioms. Some take SAS as an axiom; others don't. You may be interested in this post on my blog, putting together several answers to questions about variations between sets of axioms/postulates: Who Moved My Postulate?

As for your specific question, you'd have to prove the Law of Cosines from some specific set of geometrical axioms to see where it comes from. It would be interesting to see such a proof, which would include proving that the cosine function is well-defined, and so on. You can't just talk about "as far as I know".

Actually, now that I think about it, Euclid proved what is essentially the Law of Cosines: https://mathcs.clarku.edu/~djoyce/java/elements/bookII/propII12.html . I'd have to look closely to see whether this serves as an independent proof of SAS, and whether it is fully justified by his axioms.

where [MATH]C = \theta + x[/MATH] and [MATH]c=e+d[/MATH]
[MATH]c^2=e^2+d^2+2de[/MATH]
[MATH]c^2 = (a^2-h^2) + (b^2 - h^2) + 2ed[/MATH]
[MATH]c^2 = a^2 + b^2 - 2h^2 + 2(a\sin\theta)(b\sin x)[/MATH]
[MATH]c^2 = a^2 + b^2 - 2h^2 + 2ab\sin\theta\sin x[/MATH] and using the cosine identity [MATH]\cos(\theta+x)=\cos\theta\cos x-\sin\theta\sin x[/MATH]
[MATH]c^2 = a^2 + b^2 - 2h^2 + 2ab\cos\theta\cos x - 2ab\cos(\theta+x)[/MATH]
[MATH]c^2 = a^2 + b^2 - 2h^2 + 2(a\cos\theta)(b\cos x) - 2ab\cos C[/MATH]
[MATH]c^2 = a^2 + b^2 - 2h^2 + 2h^2 - 2ab\cos(\theta+x)[/MATH]
[MATH]c^2 = a^2 + b^2 - 2ab\cos C[/MATH]
and for a proof for the [MATH]\cos(a+b)[/MATH] identity, the next image proves the sine identity(I took the proof from Philips Lloyd, not sure if someone else found it first)



and from that and the definition of sine and cosine:

[MATH]\sin(90 - x) = \cos x[/MATH] and vice versa

[MATH]\sin((90 - a) - b) = \sin(90 - a)\cos b - \cos(90 - a)\sin b[/MATH]
[MATH]\sin(90 - (a + b)) = \cos a\cos b - \sin a\sin b[/MATH]
[MATH]\cos(a+b) = \cos a\cos b - \sin a\sin b[/MATH]and anything before that is the area of triangles formula and definitions of the sine and cosine functions so yeah.

 

[MegaMoh]

View attachment 12686

where [MATH]C = \theta + x[/MATH] and [MATH]c=e+d[/MATH]
[MATH]c^2=e^2+d^2+2de[/MATH]
[MATH]c^2 = (a^2-h^2) + (b^2 - h^2) + 2ed[/MATH]
[MATH]c^2 = a^2 + b^2 - 2h^2 + 2(a\sin\theta)(b\sin x)[/MATH]
[MATH]c^2 = a^2 + b^2 - 2h^2 + 2ab\sin\theta\sin x[/MATH] and using the cosine identity [MATH]\cos(\theta+x)=\cos\theta\cos x-\sin\theta\sin x[/MATH]
[MATH]c^2 = a^2 + b^2 - 2h^2 + 2ab\cos\theta\cos x - 2ab\cos(\theta+x)[/MATH]
[MATH]c^2 = a^2 + b^2 - 2h^2 + 2(a\cos\theta)(b\cos x) - 2ab\cos C[/MATH]
[MATH]c^2 = a^2 + b^2 - 2h^2 + 2h^2 - 2ab\cos(\theta+x)[/MATH]
[MATH]c^2 = a^2 + b^2 - 2ab\cos C[/MATH]
and for a proof for the [MATH]\cos(a+b)[/MATH] identity, the next image proves the sine identity(I took the proof from Philips Lloyd, not sure if someone else found it first)

View attachment 12687

and from that and the definition of sine and cosine:

[MATH]\sin(90 - x) = \cos x[/MATH] and vice versa

[MATH]\sin((90 - a) - b) = \sin(90 - a)\cos b - \cos(90 - a)\sin b[/MATH]
[MATH]\sin(90 - (a + b)) = \cos a\cos b - \sin a\sin b[/MATH]
[MATH]\cos(a+b) = \cos a\cos b - \sin a\sin b[/MATH]and anything before that is the area of triangles formula and definitions of the sine and cosine functions so yeah.

Edit: I saw JeffM's comment and remembered the [MATH]a^2+b^2=c^2[/MATH], which was used in these proofs, which has many proofs that do not require SAS.

 

[Dr.Peterson]

I saw JeffM's comment and remembered the [MATH]a^2+b^2=c^2[/MATH], which was used in these proofs, which has many proofs that do not require SAS.

In order to complete your proof that this does not depend on SAS, you will have to find a specific proof of Pythagoras and show that it does not depend on SAS (or some other axiom that takes its place). Once again, broad generalities (like your earlier "as far as I know") are insufficient for the kind of claim you are making.

 

[JeffM]

Edit: I saw JeffM's comment and remembered the [MATH]a^2+b^2=c^2[/MATH], which was used in these proofs, which has many proofs that do not require SAS.

You are, I think, missing the point.

Euclid claimed to prove SAS from his axioms and three earlier theorems. That claim has been shown to contain very subtle flaws that were not discovered for thousands of years. And he used SAS to prove the Pythagorean Theorem, which he then used to prove the law of cosines. It is a sequence of proofs, but the sequence is slightly flawed.

With the addition of certain axioms to Euclid, eg Hilbert's, you can fix the subtle flaws in Euclid's SAS proof. With the addition of axioms different from Hilbert's, you may be able to prove the Pythagorean Theorem without SAS, then prove the law of cosines, and finally use that to prove SAS. What are those additional axioms, and what is the sequence of proofs? Unless you show the complete sequence, you have shown nothing.

I suspect what is really bothering you is a claim that SAS must be an axiom. That claim is not true. What is true is that you cannot prove SAS from just Euclid's axioms; you need additional ones. ONE way to do that is to make SAS itself an axiom, but that is not necessary.

 

[MegaMoh]

In order to complete your proof that this does not depend on SAS, you will have to find a specific proof of Pythagoras and show that it does not depend on SAS (or some other axiom that takes its place). Once again, broad generalities (like your earlier "as far as I know") are insufficient for the kind of claim you are making.

there

this holds for any right triangle with side lengths [MATH]a[/MATH] and [MATH]b[/MATH] and hypotenuse [MATH]c[/MATH]. All it depends on is the area of a square and triangle.

You are, I think, missing the point.

Euclid claimed to prove SAS from his axioms and three earlier theorems. That claim has been shown to contain very subtle flaws that were not discovered for thousands of years. And he used SAS to prove the Pythagorean Theorem, which he then used to prove the law of cosines. It is a sequence of proofs, but the sequence is slightly flawed.

With the addition of certain axioms to Euclid, eg Hilbert's, you can fix the subtle flaws in Euclid's SAS proof. With the addition of axioms different from Hilbert's, you may be able to prove the Pythagorean Theorem without SAS, then prove the law of cosines, and finally use that to prove SAS. What are those additional axioms, and what is the sequence of proofs? Unless you show the complete sequence, you have shown nothing.

I suspect what is really bothering you is a claim that SAS must be an axiom. That claim is not true. What is true is that you cannot prove SAS from just Euclid's axioms; you need additional ones. ONE way to do that is to make SAS itself an axiom, but that is not necessary.

My point is that "one way" that says SAS is an axiom is completely unnecessary. Using any set of axioms, you could reach the simple theorems that the proofs I've shown depend on(area of a square, area of a triangle, and the definition of sine and cosine) which do not use SAS(I don't know what kind of set of axioms make it not possible to prove those simple theorem without SAS) and combining them you could see that SAS doesn't even need to be an axiom, that is my point

 

[Dr.Peterson]

What I said still applies: just waving your hand and claiming that your proof doesn't depend on SAS proves nothing. If you want to show that you can derive SAS from a particular axiom system, you have to state your axioms explicitly, and then carry out the entire process from those axioms, assuming no proofs as already having been done. I'm only repeating what JeffM said:

What are those additional axioms, and what is the sequence of proofs? Unless you show the complete sequence, you have shown nothing.

The whole point is that there are subtle flaws in Euclid's details. So you have to be looking at details, and focusing on subtlety! That's not what you're doing:

My point is that "one way" that says SAS is an axiom is completely unnecessary. Using any set of axioms, you could reach the simple theorems that the proofs I've shown depend on(area of a square, area of a triangle, and the definition of sine and cosine) which do not use SAS (I don't know what kind of set of axioms make it not possible to prove those simple theorem without SAS) and combining them you could see that SAS doesn't even need to be an axiom, that is my point

You can't really mean any set; and the only way to show what you could do is to actually do it.

As has been said, we agree that SAS doesn't have to be an axiom; but that depends on what other axioms you use, and you are refusing to think that deeply.

 

[JeffM]

Using any set of axioms, you could reach the simple theorems that the proofs I've shown depend on(area of a square, area of a triangle, and the definition of sine and cosine) which do not use SAS

PROVE it.

 

[MegaMoh]

What I said still applies: just waving your hand and claiming that your proof doesn't depend on SAS proves nothing. If you want to show that you can derive SAS from a particular axiom system, you have to state your axioms explicitly, and then carry out the entire process from those axioms, assuming no proofs as already having been done. I'm only repeating what JeffM said:


The whole point is that there are subtle flaws in Euclid's details. So you have to be looking at details, and focusing on subtlety! That's not what you're doing:

You can't really mean any set; and the only way to show what you could do is to actually do it.

As has been said, we agree that SAS doesn't have to be an axiom; but that depends on what other axioms you use, and you are refusing to think that deeply.

PROVE it.

Give me 1 set of axioms that require SAS to prove the area of a square and triangle and the definitions of sine and cosine. If there is, then your argument would make sense. Otherwise, I see no reason to ever consider SAS an axiom.

 

[JeffM]

Give me 1 set of axioms that require SAS to prove the area of a square and triangle and the definitions of sine and cosine. If there is, then your argument would make sense. Otherwise, I see no reason to ever consider SAS an axiom.

It is up to you to prove either that you can prove things about the areas of triangles and rectangles without SAS or that SAS can be derived from Euclid's axioms. You are the one making claims, but you have not listed a single axiom or proved a single theorem. Your argument is a joke.

 

[Deleted member 4993]

Can we please close this thread?!!