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In middle school, cos and sin were defined with angles, and in high school, with the length of an arc of the unit circle. But angles, are defined with cos and you need integrals to define the length of an arc! And the definition with series is a bit abstract. Is there a simple way to define cos and sin without either using advanced tools or "lying" to students?

Soc
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  • Why do you think we need integrals to define "length of arc of a circle"? It has a more or less obvious intuitive meaning which is, imo, more than enough to work with in high school for trigonometric functions. In fact, adopting some methods of teaching, we wouldn't even need the length of acircle's arc but rather only the proportiong between its length and the circle's perimeter. Of course, how do we know what's the permieter of a circle? Well, as ancient greeks, we can do approximations. But formally (univ. level), we'd need integrals. At some point one must cut and "assume" stuff. – Timbuc Feb 22 '15 at 13:06
  • So the answer to my question is "no"? – Soc Feb 22 '15 at 13:11
  • From my point of view the answer is "I've no idea", as I cannot be sure what "rigorously" means in general. – Timbuc Feb 22 '15 at 13:16
  • By "rigorously" I meant formally, without resorting to "obvious intuitive meaning".

    I don't have a problem with the way it's taught in high school, it's just that in univ, we did use cos and sin before doing integrals and series, and it bothered me a bit.

     – Soc Feb 22 '15 at 13:25
  • I think that what you did in univ was: Take the unit circle. Call $2\pi$ to a full rotation. Then by dividing in half you can define the trigs of $2\pi n/2^k$. Then extend by continuity to all angles. –  Feb 22 '15 at 13:35
  • @Soc I think the way is done in high school, namely: first defining on straight angle triangle, then expanding the definition by means of the unit (or also called "trigonomeric") circle is pretty rigorous within the limits of what's reasonable. For example, how "rigorous" is the use of point, line and etc. at the beginning of Euclidean geometry? There notions are considered "primitive" or primary and remain undefined! – Timbuc Feb 22 '15 at 13:38
  • Notice that in my comment above you don't need to claim that $2\pi$ is a length (a real number). You can take it as a symbol for the full rotation. Notice also that this doesn't prevent you from doing calculus (derivatives etc) with trigonometric functions: $\sin(2\pi x)'=2\pi\cos(2\pi x)$. You only need to "extend" the reals with the symbol $2\pi$, which we kind do in writing, until you define it as a number using, for example arc length. –  Feb 22 '15 at 13:46
  • "Rigorous" ought to mean amenable to use in a sound proof-checking algorithm. – Michael Hardy Feb 22 '15 at 13:50
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    By the way, it is not that I recommend torturing a class with the idea of $2\pi$ as a only a symbol. One can perfectly not wait until integration to define $2\pi$ as a number: One takes the lengths of inscribed and circumscribed regular polygons with $2^n$ sides on the circle. This forms nested intervals. Then show that the difference of their length tends to zero. Define $2\pi$ as the common number in between. –  Feb 22 '15 at 13:57
  • @user21870 What we did in univ was just run with the high school definition until we began doing infinite series, at which point we retroactively justified everything. But thanks for your answer. – Soc Feb 22 '15 at 14:56
  • Spivak's Calculus defines the trigonometric functions and $\pi$ using the fundamental theorem of calculus, the intermediate value theorem and other results involving inverse functions. You might want to check it out. All familiar results regarding $\sin$,$\cos$,$\tan$ and their inverses are then proved. Other results can be found in the problems. – Reveillark Feb 22 '15 at 15:06
  • I'll check it out. Thanks – Soc Feb 22 '15 at 15:35

2 Answers2

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Two comments:

  • You do not need integrals to define the length of an arc. The length is the smallest upper bound of the set of all lengths of polygonal paths moving in one direction along the arc. That is simpler than any definition involving integrals and, and the definitions involving integrals that are usually presented earliest in first- and second-year calculus and most advanced calculus courses are not rigorous. Usually they say something along the lines of $$\text{arc length}=\int_{(x_0,y_0)}^{(x_1,y_1)} \sqrt{(dx)^2+(dy)^2}.\quad\text{(This is not rigorous.)}\tag 1$$ That opens lots of cans of worms that need not be dealt with in order to define the length of an arc, including non-essential questions about differentiability, such as at which sets of points functions need to be differentiable in order that $(1)$ be valid. There are instances in which $x$ and $y$ are differentiable functions of some parameter everywhere except on some set of measure $0$ and continuous everywhere and yet $(1)$ is not valid and gives a smaller number than the actual arc length. An example is the graph of the Cantor function.
  • Yet another way of defining the sine function is as a function satisfying $f''=-f$ and $f(0)=0$ and $f'(0)=1$. This may lead to the question of how we know that such a function exists. A proof may involve a sequence of functions that converges uniformly on bounded sets to a solution.
Michael Hardy
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  • The length of an arc of a circle can be defined using “Archimedean” methods, showing that there's only one length between the inner and the outer polygonal paths. Then the arc length can be used as the definition of the angle measure. – egreg Feb 22 '15 at 14:35
  • @egreg : That certainly works in this case, when the arc is a circle. ${}\qquad{}$ – Michael Hardy Feb 22 '15 at 14:39
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By Euler's formula:

$$\sin\theta = \frac{e^{i \theta} - e^{-i \theta}}{2i} \;$$

$$\cos\theta = \frac{e^{i \theta} + e^{-i \theta}}{2} \;$$ with $\theta$ in radians.

The only things you need:

  • Exponentiation
  • Complex numbers ($i$)
  • Pi
  • The constant $e$, which can be defined using a limit.

I don't know what "advanced" tools mean because it's a subjective word. But I give my answer in hope that it is useful to you.

Edit: There is a definition of the function $e^x$ in term of a limit: $$e^x = \lim_{n \rightarrow \infty} \left(1 + \frac{x}{n}\right)^n$$

Paiam Risarki
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    How do you define exponentiation without running into the very problems raised in the initial question. The question says definitions via series are "abstract", and the same would apply to definitions of exponentiation. Besides, usually one defines $z\mapsto e^z$ without first defining $e$ and without first defining exponentiation, and then uses that function to define exponentiation in general. So I can't see that you're getting anywhere. – Michael Hardy Feb 22 '15 at 14:20
  • How would exponentiation be "abstract"? Indeed this is a vaguely-worded question, and I think that you have a different interpretation of this from me. – Paiam Risarki Feb 22 '15 at 14:22
  • It is vaguely worded, but the questioner finds definitions via infinite series too "abstract". So we have $\displaystyle\exp z=\sum_{n=0}^\infty\frac{z^n}{n!}$, and then we define $e=\exp 1$. Is that less "abstract" than $\displaystyle\cos z=\sum_{n=0}^\infty(-1)^n\frac{z^{2n}}{(2n)!}\ {}$? ${}\qquad{}$ – Michael Hardy Feb 22 '15 at 14:28
  • Okay I could have worded my question better. Let's say I simply meant without series or integral. Then I don't know how to define the exponential of an imaginary number without resorting to the exp function (therefore a series) or cos and sin, which have us running in circle – Soc Feb 22 '15 at 14:51
  • @MichaelHardy $e^x = \lim_{n \rightarrow \infty} \left(1 + \frac{x}{n}\right)^n$ – Paiam Risarki Feb 22 '15 at 15:04
  • I saw your edit. That works too. Thanks for your answer. – Soc Feb 22 '15 at 15:04