I particularly enjoyed having the infinite series model click intuitively, and seeing that the unit circle contained all possible right triangles. Why, yes, yes it does!
Your graph “Better Models of Sine” illustrating the successive series approximations of sine has an error: it indicates that sin(x) = x - x³/3! + x⁵/5! - x⁷/y! + x⁹/9! - x⁹/9! + x¹¹/11! …, which includes x⁹ twice rather than once. With as many terms, it should be sin(x) = x - x³/3! + x⁵/5! - x⁷/y! + x⁹/9! - x¹¹/11! + x¹³/13! …
Wow, thanks once again Kalid. Your explanatiosn are truly wonderful, just how do you come to such a level of knowledge and how do you manage to explain it so easily?
I wish you were my teacher.
Every article is just magic, please keep writing it’s a real relief every time you release another article.
That was a fantastic lesson. Since I left school I’ve come back to math every few years to try and remember everything I’d forgotten. The best feeling in the world (yep, even better than that one…) is the “Eureka!” moment when everything just makes sense. Your article gave me two of those, from watching Hubert move in his circle and from seeing the derivitave definition of sine and how it related to e. You have a gift for teaching and writing, thank you for sharing it.
@nschoe: Thanks for dropping by! I appreciate the kind words – I don’t think I really understand that much, it’s more my lack of understanding/satisfaction which drives me to seek simpler explanations. The notion that sine is this cyclical wave that we all see just didn’t click deeply with me, I needed something deeper. Many ideas are like that (e, imaginaries, etc.) so I start trying to find analogies that might fit better :).
I’ll definitely keep writing, appreciate the support!
@loimprevisto: You’re welcome! You got it, those Eureka moments are so incredibly fulfilling. It’s what I strive for when writing, I just want to share what clicked hoping it clicks for other people too. Thanks for sharing what aspects helped (Hubert / derivative definition), I have a project in the works to make these insight exchanges easier & more community driven :).