Here are some of my ideas:

1. Addition Formula: $\sin{x}$ and $\cos{x}$ are the unique functions satisfying:

  • $\sin(x + y) = \sin x \cos y + \cos x \sin y $

  • $\cos(x + y) = \cos x \cos y - \sin x \sin y$

  • $\sin 0 = 0\quad$ and $\quad\displaystyle{\lim_{x \rightarrow 0} \frac{\sin x }{x} = 1}$

  • $\cos 0 = 1\quad$ and $\quad\displaystyle{\lim_{x \rightarrow 0} \frac{1-\cos x}{x} = 0}$

2. Taylor Series:

  • $\displaystyle{\sin x = \sum_{n = 0}^{\infty} \frac{(-1)^n}{(2n+1)!}\;x^{2n+1}}$

  • $\displaystyle{\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!}\;x^{2n}}$

3. Differential Equations: $\sin(x)$ and $\cos(x)$ are the unique solutions to $y'' = -y$, where $\sin(0) = \cos^\prime(0) = 0$ and $\sin^\prime(0) = \cos(0) = 1$.

4. Inverse Formula: We have:

$$\begin{align} \arcsin x &= \phantom{\frac{\pi}{2} + } \int_0^x \frac{1}{\sqrt{1 - x^2}}\, dx \\[6pt] \arccos x &= \frac{\pi}{2} + \int_0^x \frac{x}{\sqrt{1 - x^2}}\, dx \end{align}$$

Then $\sin x$ is the inverse of $\arcsin x$, extended appropriately to the real line, and $\cos x$ is similar.

Question: Are there any others that you like? In particular, are there any good rigorous ones coming from the original geometric definition?

  • 2
    $\begingroup$ The most common rigorous definition is done with power series. Look at Spivak's Calculus for a rigorous construction which is based more or less on geometric ideas. All the proofs are analytical, but you can see that the ideas are motivated geometrically. $\endgroup$ – Reveillark Mar 5 '15 at 3:16
  • $\begingroup$ My answer to a related question shows a link between the geometric definitions and the power series. $\endgroup$ – Blue Mar 5 '15 at 11:48
  • $\begingroup$ I would think that a rigorous definition of these functions other than the power series would be a set of properties of $\sin$ which could be used to find each of its derivatives, allowing the construction of the power series. You could then define $\cos(x)$ as $\sin(x+\frac{\pi}4)$, and the definition of $\pi$ would be half the period of $\sin$. $\endgroup$ – u8y7541 Sep 29 '17 at 1:05
  • $\begingroup$ I consider the first approach (with a slight variation) in a similar post titled Is there a purely analytic proof of these inequalities? I wanted to know whether the limits can be proved, but it seems I'd have to assume them too. $\endgroup$ – Allawonder Jan 31 '19 at 17:40

The book Principles of Mathematical Analysis (also called Baby Rudin) by Walter Rudin, Second edition, pages 167-169, briefly develops the theory of trigonometric functions. This is after developing the theory of series of complex numbers as well as the theory of exponential and logarithmic functions, so the additional analysis can be quite brief.

Rudin defines the complex exponential function $E(z)$ by

$$E(z)=\sum_{n=0}^{\infty} \frac{z^n}{n!}$$

then defines the functions cosine and sine (which he initially calls $C(x)$ and $S(x)$) of a real variable by

$$C(x)=\frac 12[E(ix)+E(-ix)], \quad S(x)=\frac 1{2i}[E(ix)-E(-ix)]$$

He then derives all the usual properties (both trigonometric and analytic) of sine and cosine in just two pages. All his proofs are simple and clear, except for one: showing that there are positive numbers $x$ such that $C(x)=0$. That can be proved simply in other ways, so the analytic exposition of trigonometry is rigorous, brief, and clear. Along the way he also shows the usual parameterization of the unit circle and shows the usual high-school definitions of sine and cosine.

Rudin's conclusion is,

It should be stressed that we derived the basic properties of the trigonometric functions from [the definitions of $E(x)$, $C(x)$, and $S(x)$], without any appeal to the geometric notion of angle.

(I replaced references to specific equations with their meaning in this quote.)

Regarding your question "In particular, are there any good rigorous ones coming from the original geometric definition?":

The "original geometric definition" is not rigorous unless you have a good definition of the length of a circular arc as well as a good axiom system for the real numbers or an equivalent. Euclid did not provide such a system, though he tried. The explanation I gave from Baby Rudin defines geometry and trigonometry from analysis. Developing a rigorous geometry that includes all the trigonometric ideas without starting from analysis is very difficult. Even Hilbert's axioms for geometry had its problems. I have never seen a good rigorous development of trigonometry from modern formal geometry, though it must exist somewhere.

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  • $\begingroup$ Thanks for the answer! I think the easiest way to get at a rigorous notion of 'angle' is through area rather than length, but I think this essentially reproduces my suggestion 4 above. $\endgroup$ – Alex Zorn Mar 5 '15 at 21:55

My preferred definition is:

$\cos x$ and $\sin x$ are the real and imaginary parts of the exponential function $\exp(ix)$.

Since we have: $$ \begin{split} e^{ix}= \sum_{k=0}^\infty\dfrac{(ix)^k}{k!}&=1+ix+\dfrac{(ix)^2}{2!}+\dfrac{(ix)^3}{3!}+\cdots+\dfrac{(ix)^n}{n!}+\cdots\\ &=1-\dfrac{(x)^2}{2!}+\dfrac{(x)^4}{4!}-\dfrac{(x)^6}{6!}+ \cdots +i\left[ x-\dfrac{(x)^3}{3!}+\dfrac{(x)^5}{5!}+\cdots \right]\\ &= \sum_{k=0}^\infty\dfrac{(-1)^k\,x^{2k}}{(2k)!}+i \sum_{k=0}^\infty\dfrac{(-1)^k\,x^{2k+1}}{(2k+1)!}\end{split} $$ we find the usual series definitions.

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  • $\begingroup$ thanks to @Rory for edit. My ital-english is funny? $\endgroup$ – Emilio Novati Mar 5 '15 at 13:16
  • $\begingroup$ You are welcome, though I do not really deserve your thanks. At first, I thought I was editing my answer! I decided to complete the minor edits on your answer. Your English is quite good, much better than my Italian. $\endgroup$ – Rory Daulton Mar 5 '15 at 13:21

One definition that I think has a very clear geometric meaning is this: On the unit circle in the $xy$-plane, draw a line segment from the origin to a point on the circle. Call the angle that the line segment makes with the $x$-axis $t$ as shown in the picture. We define the $x$-component of the endpoint of the line segment as $\cos(t)$ and the $y$-component as $\sin(t)$.

The trig class "opposite, ajacent, hypotenuse" definition follows from this one. It is also not too difficult to derive inverse function integral definition from this. The addition formula definition is a bit more work to obtain, but there are geometric proofs of all of the facts listed.

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  • $\begingroup$ This is a good intuitive definition of sine and cosine, but did you read the other answers? This definition was already mentioned, along with arguments why it is not rigorous. Do you have a response to those arguments? $\endgroup$ – David K Aug 7 at 13:18
  • $\begingroup$ @DavidK I assume you are referring to the section at the end of Rory Daulton's answer. It seemed unclear to me that this definition fit that category, because I did not see where I used arc length. I now think I understand the argument: you need length of a circular arc to define angle? In this case, as he says, if we have both real numbers and length of a circular arc, then there shouldn't be a problem with the definition. $\endgroup$ – Math2718 Aug 7 at 14:45

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