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This is a pretty open ended question and I apologize, in advance, if this is not the place for it. But what do you recommend should be given the title of the Fundamental Theorem of Trigonometry and why? Should we have to restrict ourselves to the planar case... I think so.

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    $\begingroup$ $\sin^2x+\cos^2x=1 ?$ $\endgroup$ – Lucian Nov 19 '13 at 1:05
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    $\begingroup$ Unless you're feeling particularly stingy with your mouse clicks, you can give a "check" to comments... $\endgroup$ – The Chaz 2.0 Nov 19 '13 at 1:23
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    $\begingroup$ Fundamental theorems are never exotic. They are plain dull and obvious. Take the fundamental theorems of analysis or algebra, for instance. $\endgroup$ – Lucian Nov 19 '13 at 2:12
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    $\begingroup$ I didn't think FT of algebra was, nor the result for Riemannian Geometry. $\endgroup$ – Squirtle Nov 20 '13 at 2:35
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    $\begingroup$ The sum of the angles is $180^\circ$. $\endgroup$ – Christian Blatter Sep 30 '14 at 11:10
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$$\boxed{\sin^2x+\cos^2x=1.}$$

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    $\begingroup$ It was recognized by the 1800s that the angle difference formula for cosine $$\cos(a-b) = \cos a \cos b + \sin a \sin b$$ is fundamental in the sense that the other relations of trigonometry can be derived from it using a functional equation approach. See Aczel on this. Also, though people often believe the contrary, $\cos^2 x + \sin^2 x = 1$ for acute angles can be derived from the angle difference formulas independently of Pythagoras. (See my article in Forum Geometricorum.) $\endgroup$ – Jason Zimba Mar 27 '14 at 14:50
  • $\begingroup$ Related:i.stack.imgur.com/MNFcV.png $\endgroup$ – Guy Mar 27 '14 at 15:08
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Fundamental theorem, imho, would be:

A magnifying glass that increases the size of an object $k$ times:
1) Doesn't change angles
2) Increases length by a factor of $k$

From this you can (informally) derive the existence of sine, cosine, $\pi$, the $k^2$ increase in area, figure out problems of similar triangles, etc.

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    $\begingroup$ I'm afraid that's more on the level of axioms rather than theorems. $\endgroup$ – Lucian Nov 19 '13 at 2:12
  • $\begingroup$ I completely agree $\endgroup$ – Squirtle Nov 19 '13 at 3:35
  • $\begingroup$ I like this, but what is the definition of size? I think it should be 1) implies 2) or perhaps even 2) implies 1) $\endgroup$ – JP McCarthy Nov 19 '13 at 11:36
  • $\begingroup$ I really meant this informally. Trig is a middle school / high school level class where I'm from usually, so I wanted to suggest an observation that students could make which leads to many trig results. I doubt anyone would use this as a theorem, much less as an axiom, in any formal system. $\endgroup$ – DanielV Nov 19 '13 at 15:20
  • $\begingroup$ @JpMcCarthy If I was (unlikely) to try to formalize this, I wouldn't define length or angles. I would leave them undefined but with constraints, similar to how peano arithmetic does with zero/successor. $\endgroup$ – DanielV Nov 19 '13 at 15:23
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Answer: The Fundamental Theorem of Trigonometry is

In a unit circle, an arc of length $2x$ stands on a chord of length $2sin(x)$.

Source: Goodstein's Mathematical Analysis

Argument: This theorem connects the geometric definition of the trig functions with the analytic definition of the trig functions.

Proof: The points $(sin(\alpha), cos(\alpha))$ and $(-sin(\alpha), cos(\alpha))$ with $0 \leq \alpha \leq \frac{1}{2}\pi$ are the endpoints of a chord on the unit circle $x^2+y^2=1$ having length $2sin(\alpha)$. The length of the arc joining them is $$ \int_{-sin(\alpha)}^{\sin(\alpha)} \sqrt{1 + \left(\frac{dy}{dx}\right)^2}dx = \left[arcsin(x)\right]^{sin(\alpha)}_{-sin(\alpha)} =2 \alpha $$

Remark: The argument that the integral is equal to $2\alpha$ uses the definition of sine as the limit of a power series (or whatever analytic definition you feel is most appropriate), but the coordinates come from the definition of sine as the x-coordinate of the point of intersection of a line through the center of a unit circle.

Note: This proof can be found on page 166-167 of Goodstein's Mathematical Analysis.

(Additionally one can see this theorem as fundamental in that it connects the ancient development of trigonometry to our modern use of the subject: https://en.m.wikipedia.org/wiki/Chord_(geometry) ).

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The identity $\sin^2x+\cos^2x=1$ comes from Pythagoras.

I think the fact that $\sin x$ and $\cos x$ (for a right-angled triangle) are well defined at all is the fundamental theorem.

Fundamental Theorem of Trigonometry The ratio between corresponding sides of similar triangles are equal.

Edit: Daniel V has a similar idea.

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  • $\begingroup$ This is called the Fundamental Theorem of Similarity. Though since one way of proving Pythagoras is precisely through similarity of (straight-edged) triangles, it makes a lot of sense. $\endgroup$ – Lucian Nov 19 '13 at 23:01
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It was recognized by the 1800s that the angle difference formula for cosine $$\cos(a-b) = \cos a \cos b + \sin a \sin b$$ is fundamental in the sense that the other relations of trigonometry can be derived from it using a functional equation approach. See Aczel on this.

Also, though people often believe the contrary, $\cos^2 x + \sin^2 x = 1$ for acute angles can be derived from the angle difference formulas independently of Pythagoras. (See my article in Forum Geometricorum.)

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    $\begingroup$ i.stack.imgur.com/MNFcV.png derived without pythagoras. no angle difference either. $\endgroup$ – Guy Mar 27 '14 at 14:52
  • $\begingroup$ @Sabyasachi fair enough! I had set myself the problem of devising a proof that did not rely on a pre-existing proof of the Pythagorean theorem. An intellectual exercise. Anyway my main point is the one from Aczel, which makes the top answer to this post less satisfying I think. $\endgroup$ – Jason Zimba Mar 27 '14 at 14:58
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    $\begingroup$ @Sabyasachi At' the risk of detracting even further from my main point, in case interesting the proof is here: forumgeom.fau.edu/FG2009volume9/FG200925.pdf. It can be rendered diagramatically too - in fact, in many ways since there is a slack parameter. Again in case interesting: youtube.com/watch?v=TOsXoRTP0Yc. $\endgroup$ – Jason Zimba Mar 27 '14 at 15:02
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    $\begingroup$ if you want to, you can even proof pythagoras from Euler's identity. The order in which the proofs where discovered historically, hardly matters imo. $\endgroup$ – Guy Mar 27 '14 at 15:04
  • $\begingroup$ I think that a "fundamental" theorem would be something that is immediately obvious as to why it is true. My picture shows that for $\sin^2(x)+\cos^2(x)$. Although if you google, you can find nice proofs for your identity as well. $\endgroup$ – Guy Mar 27 '14 at 15:06
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$e^{i\theta}=\cos \theta + i \sin \theta$.

You can derive a bunch of the relations from here!

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  • $\begingroup$ This seems more like a result from complex analysis than trig. $\endgroup$ – Squirtle Nov 19 '13 at 3:40
  • $\begingroup$ @Squirtle: Euler's formula may stem from complex analysis but $\text{cis}\left(\theta\right)$ is trig nonetheless. $\endgroup$ – Nick Sep 30 '14 at 15:29
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My attempt: a right triangle with unit hypothenuse and angle $x$ has base $\cos x$.

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  • $\begingroup$ and an altitude $\sin x$ $\endgroup$ – Nick Sep 30 '14 at 12:21
  • $\begingroup$ @Nick: this is implicit, as the Pythagoras theorem establishes it independently. $\endgroup$ – Yves Daoust Sep 30 '14 at 13:36
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$$\frac{\sin x}{\cos x} = \tan x$$ $$sin^2 x + cos^2x = 1$$

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    $\begingroup$ The first equality is the definition of $\tan$. $\endgroup$ – Martín-Blas Pérez Pinilla Mar 27 '14 at 15:23
  • $\begingroup$ Lol sinx/cosx=tanx ... $\endgroup$ – Cloud JR Aug 6 '18 at 19:57

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