How to define trigonometric functions analytically and elementary

Question

Let's say I want to study real analysis in a self-contained fashion by starting with the axioms of the real numbers. This is because I already know the basic concepts of set theory and high school calculus.

I have already defined the concepts of the exponential and logarithmic functions by using the upper bound property. I also defined the decimal representation of numbers, again by using the upper bound property.

My problem is how to define trigonometric functions before I even start the study of derivatives, integrals or power series. In this I'm thinking of a definition analytically, meaning without appealing to geometry other than intuitively.

The reason I'm trying to do this is because in the study of derivatives or integrals they use a lot of examples including real functions without they being previously defined, which makes me feel a little uncomfortable. One way to get around this is probably by just thinking of those functions axiomatically, meaning they will be defined at some point later and meanwhile they have certain properties that I can use to find derivatives and integrals, but still I wanted to know the elementary way to accomplish this.

Answer 1

A perfectly analytical way to define trigonometric functions is via their power series expansion, albeit this definition is not really intuitive if you are restricting to the real line. But if you are not, then you could obtain the series for sine and cosine by separating the power series expansion of the complex exponential function into real/imaginary parts and then referring to Euler's formula.

Answer 2

I would start with either Euler's formula or Descartes formula:

$$e^{ix}=\cos(x)+i\sin(x)$$

Where $e$ is Euler's constant.

$$(a+bi)^n=|a+bi|^n[\cos(n\theta)+i\sin(n\theta)]$$

Where $\theta$ is the angle of $a+bi$ when graphed on the complex plane with respect to the positive real axis.

You can derive most of your trig formulas from these, mostly from the first formula, and if you try to combine the two, you should get how to turn a complex number from Cartesian coordinates to polar.

You can also define in the following manner:

$$\frac{\sin(x)}x=\left(1-\frac{x^2}{1^2\pi^2}\right)\left(1-\frac{x^2}{2^2\pi^2}\right)\left(1-\frac{x^2}{3^2\pi^2}\right)\dots$$

which should converge (slowly) for $x\in\mathbb C$

Answer 3

I don't think there is a way to introduce $\sin x, \cos x$ using real analysis (i.e. without any use of complex numbers) in a manner which avoids differential/integral calculus/infinite series/infinite product. There is no way to introduce them via upper bound property. The reason is that the relation between real number $x$ and real number $\sin x$ is very very very (!) indirect.

The simplest approach to circular functions $\sin x, \cos x$ is of course via the circle (that justifies their name). But such an approach requires us to establish either of the following facts:

1) An arc of a circle has a length

2) A sector of a circle has an area

Both these facts cannot be established in serious manner without using integrals (existence of arc length is possible without integrals just using upper bound property but some of the required properties of arc length do need the use of integrals).

Answer 4

One standard way to define the exponentiation (Rudin does this in some early exercises) is to first define it for integer powers, then extend to rational numbers by define $a^{m/n}$ to be the unique positive $n$th root of $a^m$, and then extend to irrational numbers by continuity. Of course, there are an annoying number of details to check, all of which Rudin lays out in the exercises. Essentially you're defining $a^x$ as the unique continuous function which obeys the functional equation $a^{x+y}=a^xa^y$ with $a^1=a$.

In principle, one could probably do the same thing for $\cos$ and $\sin$. You start by defining $\cos(\pi/2)=0$ and $\sin(\pi/2)=1$. Then you can get $\sin (\pi/2^n)$ and $\cos(\pi/2^n)$ by repeated use of half-angle formulas, and so $\sin(d\pi)$ and $\cos(d\pi)$ for $d$ a dyadic rational in $(0,1/2)$ by using the angle addition formulas. Then you extend to all $x \in [0,\pi/2]$ by continuity, and to all $x \in \Bbb{R}$ by periodicity. Basically you are defining $\sin$ and $\cos$ to be the unique continuous functions which obey the angle addition formulas and have the right values at $\pi/2$.

In practice, the details are likely to be even more horrible than they are for exponentials, and this would be ridiculously unmotivated to anyone who wasn't inclined to believe in the existence of $\sin$ and $\cos$ for some other (probably higher-level or much less rigorous) reason.

Answer 5

The trigonometric functions are: $\sin,\cos,\tan,\cot,\sec,\csc$. They are elementary standard functions.

$\sin,\cos,\tan,\cot$ and $i$ are introduced already in scool.

The trigonometric functions are rational functions of $e^{ix}$.

The elementary functions are generated from their variable by applying finite numbers of algebraic functions, $\exp$ and/or $\ln$.