What is the Fundamental Theorem of Trigonometry?

Question

Historically the fundamental theorem of trigonometry has been:

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

Sadly, I rarely if ever see mention of anything "being" the fundamental theorem of trigonometry.

Don't think "fundamental theorem of trigonometry" is something I've ever heard referenced. Got a source? Moreover, I don't think the "fundamental theorem" of anything tends to change. — AJY, Mar 02 '16 at 17:14
Source: Goodsteins's Mathematical Analysis. Argument: This fact proves that the analytic definition of $sin(x)$ can be identified with its geometric definition. — John, Mar 02 '16 at 17:19
I think that may be peculiar to that text, or it is a mostly antiquated term. — AJY, Mar 02 '16 at 17:20
No. But I've never heard the phrase, and I'm not sure what modern disciplines would use it. That's not to say it isn't "a thing", but I suspect it's not a widely used phrase. — AJY, Mar 02 '16 at 17:32
@geodude Could you explain why that is the fundamental theorem of geometry in the same way that the statement I gave is? — John, Mar 02 '16 at 17:34
@AJY Any modern discipline which uses a calculator to evaluate the sine of an angle uses the fundamental theorem of geometry to say that what has been calculated is in fact related to a geometric figure. — John, Mar 02 '16 at 17:36
I think that what is being confused here is that there is a huge difference between an equality between quantities like cos(x) and sin(x), and one between two different definitions of sin(x) (the analytic sum, and the geometric proportion). — John, Mar 02 '16 at 17:38
I might say that: The fundamental theorem of trigonometry is just as important and profound as the more familiar Fundamental Theorem of Calculus which connects the geometric area to an algebraic difference between function values. — John, Mar 02 '16 at 17:40
@AJY My question is not "Is 'The Fundamental Theorem of Trigonometry' a widely used phrase." but rather "What is the fundamental theorem of trigonometry?" — John, Mar 02 '16 at 17:43
My apologies. If there could be said to be one, I'd say it would be either the Pythagorean identity @geodude mentioned, or $e^{i x} = \cos(x) + i \sin(x)$. — AJY, Mar 02 '16 at 17:45
@AJY What argument supports your saying that either the pythagorean identity or the "Eulerian Identity" are the fundamental theorems of geometry? — John, Mar 02 '16 at 17:47
For trig, I'd say the latter because it provides a very convenient way to derive trig identities for sums of angles, multiples of angles, etc. — AJY, Mar 02 '16 at 17:48
@AJY I agree with you that both of those algebraic identities are very useful for deriving algebraic identities that relate the trigonometric functions to one another, but they do not, on their own, demonstrate that the geometric definitions of the trig functions are identical to the analytic definitions (the geometric definition is a ratio, and the analytic is often a power series). — John, Mar 02 '16 at 17:52
@AJY As I previously argued, this is as fundamental as the fundamental theorem of calculus in that it links two wildly different definitions to the same quantity. — John, Mar 02 '16 at 17:56
Thank you @Lucian I have just posted a complete answer on that thread: http://math.stackexchange.com/a/1680331/314739 — John, Mar 02 '16 at 18:35

score 0 · Answer 1 · edited Mar 02 '16 at 21:42

The closest answer to a “fundamental theorem of trigonometry” that I could find is the following, from Wikipedia. Hope this helps. $\sin^2{\theta}+\cos^2{\theta}=1$ (note that $\sin^2{\theta}$ is the same as $(\sin{θ})^2$). This is sometimes called the “fundamental Pythagorean trigonometric identity”). It follows from the Pythagorean theorem, and says that if the length of a right triangle’s hypotenuse is 1, then the length of either of the legs is the sine of the opposite angle and the cosine of the adjacent acute angle.

score 0 · Answer 2 · answered Mar 02 '16 at 18:37

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 of sine you feel is most appropriate), but the coordinates on the unit circle uses 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.

What is the Fundamental Theorem of Trigonometry?

2 Answers2