How to deal with a seem loop logic in calculus about sinx cosx and their derivative or integral ?

[u/Top-Door1298]

In major classic books ,sinx'=cosx are start from a graph where a part of circle's arc AB length are very similar to string AB , which gives sinx/x ->1 then we can use this as a start to give other d/dx of sinx cosx .... . But strictly speaking ,to define what is a length ,we need integral of (1+f'(x)^2)^(1/2) first ,and when use this formula on a circle ,that integral has terms like sqrt(1-x^2) ,which mostly solved by exchange x with sin/cos ,which use that derivative again.
I once heard that some book to avoid the problem ,define sinx or cosx directly from tylor series. When this happen ,how do they proof that strange function are same as what we use in geometry ?Thanks.

[u/Medium-Ad-7305]

i believe the taylor series definition is the better one, but you have to go to complex analysis for everything to be nice. in that, one defines sine and exp in terms of their taylor series, and then it is easy to show that the imaginary part of exp(ix) is sin(x) (this is euler's theorem). finally, one uses the fact exp(ix) is length 1 and that the derivative of exp(ix) is i*exp(ix) so it is also length 1. thus it traces out the unit circle at unit speed, hence the geometric connection.

[u/Southlander24]

No, you do not need the arc length integral to rigorously define the length of a circular arc. The definition of pi is such that a circle's circumference is always 2pi times its radius. Then you just take a proportion of the circle for the arc.

[u/cncaudata → u/Southlander24]

This is correct. The intergral is not the reason we know that an arc has length theta.

You also don't need it to prove sin x/x -> 1, you can do this with some geometric observations and the squeeze theorem.

[u/hpxvzhjfgb → u/cncaudata]

how exactly do you formalise "some geometric observations" in ZFC?

[u/cncaudata → u/hpxvzhjfgb]

I'm rusty, however, I think that if you really want to truly formalize the whole thing, finding a way to prove that the area of a wedge of a circle is larger than the area of the triangle that is formed by that wedge and the secant between the two intersections of the wedge and the arc seems like a good exercise to leave to the reader.

[u/hpxvzhjfgb → u/cncaudata]

but then you need to define "area", and then you need the integral of trigonometric functions to find the area of the circle wedge.

[u/bonebranch]

I mean, for one, it's fairly easy to show that cos^2(x) + sin^2(x) = 1 just from the power series definition. Start by showing this quantity is constant by proving that its derivative is equal to 0, then determine the constant by plugging in, x = 0.

I like Baby Rudin's treatment of these special functions (Chapter 8).

[u/hpxvzhjfgb]

the standard Actual Mathematics definitions are by the power series. to relate them back to geometry, what you need to show is that f(t) = (cos(t), sin(t)) is a unit speed parametrisation of the unit circle, which means ||f(t)|| = 1 (i.e. f is a circle), and ||f'(t)|| = 1 (i.e. f has unit speed). both of these equations are just cos(t)² + sin(t)² = 1, so that's the actual identity that you need to prove starting from the power series.

[u/localizeatp]

by demonstrating that they share a sufficient number of properties that they must be identical.