Unit circle definition is axiomatic?

Question

The unit circle definition of sin and cos state that for a point (x,y) on the unit circle x = cosA and y = sinA. Is this definition axiomatic or is it derived from somewhere ?

Answer 1

By definition $\cos \alpha$ and $\sin \alpha$ are the coordinates $(x,y)$ of a point on the unit circle centered at the origin such that the ray OP form an angle $\alpha$ with the $x$ axis, indeed

$$x^2+y^2=1 \implies \cos^2 \alpha + sin^2 \alpha =1$$

Answer 2

Whatever you want. What one calls a "definition" or "axiom" or "derived" is purely a question of pedagogy; they aren't intrinsic features of the statements themselves.

Considering the statement

The point that makes an angle $\alpha$ with the positive $x$-axis is given by $(\cos(\alpha), \sin(\alpha))$

In practice you see all three of the following in various formulations:

• This is the definition of $\cos$ and $\sin$.
• This is the definition of $\alpha$
• $\cos, \sin, \alpha$ have other definitions and this is a statement you prove

Answer 3

Referring to Gimusi's drawing:

Consider the parametrization:

$f : [0,2π) \subset \mathbb{R} \rightarrow $

$C:=${$(x,y)| x^2+y^2=1$} $\subset \mathbb{R^2},$

$t \rightarrow (\cos t, \sin t),$

is continuous and bijective.

Note : The inverse function $f^{-1}$ is not continuos at $(1,0)$.