How can you rigorously define trigonometric functions without series or integrals? 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?
 A: 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.

A: 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:
$$e^x = \lim_{n \rightarrow \infty} \left(1 + \frac{x}{n}\right)^n$$
