Question about trigonometry proofs

[CuriousBanker]

1) Are there any proofs that sine cosine and tangent are the same in all similar triangles? Like, that sine 30 degrees is the same no matter what the lengths are? Or is it just an axiom?

2) Is there a proof for the actual value for the sine cosine and tangent ratios (besides for common ones like 30-60-90). Like is there a proof that sine of 38 degrees is 0.616, or is it just that there are tables for all sine measurements that were created by observation?

Thanks in advance

 

[arildno]

1. IF two objects, say two triangles (T) og (t) are similar, THEN a) it is meaningful to say that sides l_1, l_2, l_3 "corresponds to L_1, L_2, L_3 and b) there exists a constant k, so that we have:
l_1=k*L_1
l_2 =k*L_2
l_3=k*L_3
, where capital letters are used for one of the triangles, small letters for the other.
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2. Yes, such proofs exist.
------------------------
Given this:
If we know look on RATIOS, we have that, for example:
L_1/L_2=l_1/l_2, since the common k-factor in numerator and denominator on RHS disappears.

Calling then such a ratio for, for example, "sine" should answer your question.
-----------------------------
It is typically placed in the definition of geometrical "similarity" between two objects that such a constant "k" exists, and vice versa (that is: can you find such a "k" that these relationships between all lengths in the two objects, then the two objects are "similar). However, that doesn't mean it is always easy to determine whether two objects are actually similar...

 

[SteamKing]

1. Sines, cosines, and tangents are all expressed as ratios of the various sides of a given right triangle. As long as the acute angle of the triangles are the same, these ratios will be the same. To invalidate this, you must overthrow the Pythagorean Theorem.

2. Yes, there are proofs. I don't know how you would 'observe' the value of the sine of 38 degrees. You know, trigonometry is actual math, with proofs and formulas and theorems and whatnot. It wouldn't hurt to pick up a book, or even surf the web for more info.

 

[CuriousBanker]

1. Sines, cosines, and tangents are all expressed as ratios of the various sides of a given right triangle. As long as the acute angle of the triangles are the same, these ratios will be the same. To invalidate this, you must overthrow the Pythagorean Theorem.

2. Yes, there are proofs. I don't know how you would 'observe' the value of the sine of 38 degrees. You know, trigonometry is actual math, with proofs and formulas and theorems and whatnot. It wouldn't hurt to pick up a book, or even surf the web for more info.

I have two trigonometry books, none of them showed any proofs for the sines of various degrees...also tried surfing web first, didnt find anything, so I posted here...

I was not implying that trigonometry is not real math. But you can't prove that 1 + 1 = 2...2 is DEFINED as 1+1. I was wondering if these ratios were just defined or proven, that's all

 

[CuriousBanker]

1. IF two objects, say two triangles (T) og (t) are similar, THEN a) it is meaningful to say that sides l_1, l_2, l_3 "corresponds to L_1, L_2, L_3 and b) there exists a constant k, so that we have:
l_1=k*L_1
l_2 =k*L_2
l_3=k*L_3
, where capital letters are used for one of the triangles, small letters for the other.
----------------------
2. Yes, such proofs exist.
------------------------
Given this:
If we know look on RATIOS, we have that, for example:
L_1/L_2=l_1/l_2, since the common k-factor in numerator and denominator on RHS disappears.

Calling then such a ratio for, for example, "sine" should answer your question.
-----------------------------
It is typically placed in the definition of geometrical "similarity" between two objects that such a constant "k" exists, and vice versa (that is: can you find such a "k" that these relationships between all lengths in the two objects, then the two objects are "similar). However, that doesn't mean it is always easy to determine whether two objects are actually similar...

Part 1 makes sense, thanks

I'm not quite sure I understand what you mean by part two. How do I prove that sine of 38 degrees is 0.616?

 

[brocks]

How do I prove that sine of 38 degrees is 0.616?

Actually, that would be a good exercise for you.

You start with the fact that you know exactly what the sines are for 30, 45, and 60 degrees, because you know that the sides are 1:1 or 1:2.

Then study the proofs for the formulae that give the sines of A+B, A-B, A/2, and A*2 until you understand them well enough to be convinced that nobody is trying to slip one past you.

Then just start adding, subtracting, doubling, and halving angles. If you know the sine of 30, and the sine of 45, then you can get the sines of 75, and 15. If you know the sine of 15, then you can use the half angle formula get the sine of 7.5, and 3.75, and 1.875, and .9375, and .4675, etc.

So you can get the sine of (30 + 7.5), which is the same as sine (75/2), in two different and fairly easy ways, and then you can get the sine of (37.5 + .4675), and so on, until you have it as close to sine 38 as you want.

That's pretty much how the Greeks did it, two thousand years ago. Since then, we have things like the Taylor series expansion that make it much easier, but I still think the above would be a good exercise for you.

 

[CuriousBanker]

Ah, that makes so much sense, so obvious, don't know how I didn't understand..thank you so much. I thought that the arabs simply measured the sides of one triangle with 38 degrees and said "well the ratios appear to be 0.616 so that must be what it is for all other sine 38"..

 

[SteamKing]

But you can't prove that 1 + 1 = 2...2 is DEFINED as 1+1.

I don't believe that is the case. Whitehead and Russell used a great deal of writing and logical gymnastics to prove that 1+1 = 2 in their Principia Mathematica (1913). Here is a page of interest:

http://quod.lib.umich.edu/cgi/t/tex...3201.0001.001&frm=frameset&view=image&seq=401

Again for basic trigonometry, this article is helpful:

http://en.wikipedia.org/wiki/Trigonometry

The basics are laid out in the first few paragraphs (with pitchers n' ev'rythin'.)

BTW, I'm always suspicious when people say, "I searched on the internet and I couldn't find anything!"

 

[CuriousBanker]

Well what should I have searched? I tried various ways of phrasing "proof of sine ratios" and I constantly got nothing but proof of trig identities, but not how to derive sine of 38, sine of 42, etc

 

[arildno]

"I don't believe that is the case. "

It is.

The fact that W&R defines other fundamental quantities so that a concept "2" is developed independent of a concept "1" and the operation "addition", and then goes on to prove that 1+1=2 does NOT mean that it is somehow logically indefensible to DEFINE 2=1+1

However, the latter procedure is unsuitable for deriving the set concepts W&R worked with..

 

[SteamKing]

Well what should I have searched? I tried various ways of phrasing "proof of sine ratios" and I constantly got nothing but proof of trig identities, but not how to derive sine of 38, sine of 42, etc

That's why you start with the broad topic like 'trigonometry' first. In it, you see stated very clearly the definitions of the various trig functions, and definitions generally require no proof. Now, once this is established, you can delve deeper into the subject. The trig functions are also established in terms of certain infinite series, but these series probably will not be covered in 'trigonometry'. Such topics are reserved for subjects like 'analysis' or 'calculus'.

BTW, as has been suggested in an earlier post, the trig functions of angles like 38 degrees or whatnot can be established from a triangle like a 30-60-90 by application of the trig identities you encountered.

I'm sorry you have been laboring under a misconception about trigonometry, but we are attempting to show you how these topics are defined and how you can find out more about them on your own.

 

[verty]

1) Are there any proofs that sine cosine and tangent are the same in all similar triangles? Like, that sine 30 degrees is the same no matter what the lengths are? Or is it just an axiom?

2) Is there a proof for the actual value for the sine cosine and tangent ratios (besides for common ones like 30-60-90). Like is there a proof that sine of 38 degrees is 0.616, or is it just that there are tables for all sine measurements that were created by observation?

Thanks in advance

Without ignoring what other people have said above, I wanted to mention that you can take any vector and break it into components. If you believe that, then scaling the components by the same amount scales the vector by that same amount, Pythagoras's theorem is the proof of this.

Let $V = a \vec{i} + b\vec{j}$, then $2V = 2a \vec{i} + 2b\vec{j}$, and so on. And the components are the sides of a triangle with hypotenuse |V|. This should be enough to satisfy 1).

If you don't believe that any vector can be broken up into components, well you draw a horizontal line from the start of the vector, you draw a vertical line from the end of the vector, and these two lines define the components of the vector. From their intersection, the distance to each end point of the vector is a component.