MHB What are the axioms underlying the definition of sine?

[roni1]

My class is difficult to teach, but I have a question that I think that I share the forum and if you give nice ideas it can be helpful.
This is my last question of axioms and so on because I don't want to be as a mathematic cranck.
So, this is my last question that deal with it.
What I need to answer to the question: "What are the axioms of sinus definition?"
Any ideas?

 

[I like Serena]

My class is difficult to teach, but I have a question that I think that I share the forum and if you give nice ideas it can be helpful.
This is my last question of axioms and so on because I don't want to be as a mathematic cranck.
So, this is my last question that deal with it.
What I need to answer to the question: "What are the axioms of sinus definition?"
Any ideas?

Hey roni,

Classicaly, the sine is defined through the axioms (also known as postulates) of Euclidean Geometry, which is what is needed to define a right-angled triangle. And we also need the axioms of the Field of the real numbers, since the sine is defined as the quotient of 2 real numbers.

However, the sine has a number of equivalent definitions.
One of those is:
$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots $$
With this definition it suffices to just have the axioms of a Field.
And the same definition applies both to the field of the real numbers and to the field of the complex numbers.
So this definition is more general than the geometric definition and requires fewer axioms.

For the record, there are 3 types of statements in mathematics:

1. Axioms or Postulates.
2. Definitions.
3. Propositions, Theorems, Lemmas, Corollaries (all require a Proof).

The axioms or postulates are the assumptions we make that define the framework of what follows.
Definitions define terms or symbols that we will then use afterwards.
Propositions, theorems, lemmas, and corollaries are statements that follow from the axioms and definitions. They are accompanied by a Proof to prove that they are actually true. Each of these terms is used interchangeably and formally they all mean the same thing. It's a matter of preference which one is used.

 

[S.G. Janssens]

My class is difficult to teach, but I have a question that I think that I share the forum and if you give nice ideas it can be helpful.

Still I am curious about what class you are teaching. Is it introductory analysis?

This is my last question of axioms and so on because I don't want to be as a mathematic cranck.

Asking about axioms does not make you a crank.

So, this is my last question that deal with it.
What I need to answer to the question: "What are the axioms of sinus definition?"
Any ideas?

It is my impression that some of the "axioms" you are asking about are really definitions of minimal structures (such as the algebraic structure "field") underlying the concept of interest (such as "integral" or "sine".)

In addition to the answers by ILS in post #2, I would like to offer my favorite one. Namely, given the system of differential equations
\[
\left\{
\begin{aligned}
\dot{u} &= v,\\
\dot{v} &= -u,\\
\end{aligned}
\right.
\]
with initial values $u(0) = 0, v(0) = 1$, we denote the global solution $(u, v) : \mathbb{R} \to \mathbb{R}^2$ and then we define $\sin := u$ and $\cos := v$.

If you apply Picard iteration to solve the above differential equation, then for $u$ you obtain precisely the power series from post #2, while for $v$ you obtain the familiar power series for $\cos$.

(Credits to Joost Hulshof (VU, Amsterdam) for showing this once in his Analysis I course.)