Re: sin x / x tends to 1...

Lee Rudolph wrote:

> By the definition (which most of the participants in this thread
> are using, most of the time) of sine and cosine that says "the
> point on the unit circle whose angular coordinate equals x has
> Cartesian coordinates (cos(x),sin(x)), it is tautologous that
> x \mapsto (cos(x),sin(x)) is a parametrization of the unit
> circle (with, say, domain some interval of real numbers of
> length at least 2\pi). The alternative parametrizations of
> various semicircles, such as t \mapsto (t,(1-t^2)^{1/2}) for
> the upper semicircle, shows that the classic, geometrically
> defined "tangent line" to the circle at any given point is
> also equal to the "tangent line" as defined by calculus at
> the corresponding point of that, and therefore of any,
> smooth parametrization.

Can you expand on this?
Thanks in advance

> From this it follows that, *if you believe* that sin has a
> (non-zero) derivative *at all*, at any point, then its
> derivative is cos. Can that belief be justified without
> assuming the limit under consideration? I suspect it can,
> by further geometric argument using the helix
> parametrized by x \mapsto (cos(x), sin(x), x).


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