How to define trigonometric functions analytically and elementary

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.

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.

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).