Thank you!!!
This is theeeeeee best explanation I have ever had, even after taking two semesters of Calculus!!!
To all the other great examples of periodic motion, may I add the trampoline? In my mind, I’ve been having fun bouncing from pi to pi.
You are rocking and adding new dimensions to maths
Let me see if I have this right. Sin(x) is a position function. Sin(x)=x at first, a linear relationship, because there is little or no resistance around the neutral position. But then as the distance increases, so does the resistance, and a restoring force must be added to x, which alters the position. This restoring force, -x, is negative acceleration, a pull back towards the neutral position. We know that acceleration is the second derivative of the position function. Thus, to determine the new position, we double integrate -x to arrive at -x^3/3! and add it to the series. However, this being the inexact world of harmonic motion, the new term overshoots the mark and requires a correction. So we then double integrate -x^3/3!, being sure to cancel the negatives first, which gives us x^5/5!, which also overshoots true sine and must be corrected with another term, and so on.
This is the most dynamic view of both sine and the Taylor series that I’ve ever seen.
Hi Tim! Yes, that’s exactly it. As the restoring force slows you down (changing your distance by -x^3/3!), it means you aren’t going along your original “x” trajectory any more, and should not feel the full brunt of -x^3/3! – you get a bonus of x^5/5!.
But, that bonus means you’re going further along than you expected, so feel an extra restoring force of -x^7/7!. And the cycle keeps going :). Basically, the longer you want to model sine, the more levels of restoring forces you need to stay accurate [and if we’re on a very small timescale, we just assume sin(x) = x and ignore the impact of even the first restoring force].
Glad you like the perspective, I need to see/visualize things to make sure they’re really clicking.
Your clarity of thought is a gift. I teach college and often draw upon your analogies. Thank you for sharing your insights generously.
Great article! Thanks a lot for sharing your insights!
I remember another mnemonic: Some People Have, Curly Brown Hair, Through Proper Brushes: Sine=Perpendicular/Hypotenuse, Cos=Base/Hypotenuse, Tan=Perpendicular/Base. I like it better 
This brought me to tears
WOW …amazing! Good job
thank you very much sir
Hi Dev, great question. If you have a circular path, the phase angle (aka phase offset) is where the path starts. Instead of starting at 0 degrees and spinning around, the pattern might start at 90 degrees (top of the circle) and spin around from there. There’s a bit more in this article: http://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/
This is wonderful! I’m so happy to finally have some independent concept of what sine is! And even if it doesn’t make Trig easier it does make a lot more logical sense now. Thank you for making my day=D
Great explanation. Easy to understand.
Can you explain what is phase angle or phase ofset (it cofused me when I was learning about deriving the progressive wave equation in physics)
Seriously i have no words man. . I have always hated math though i scored centum in my schools coz of sin and cos wondering what the hell do they mean “Really” . Just happened to read this and its like some mysterious secret unvieling before my eyes. Thanks a lot . Wish a i had a teacher like you. Share more of ur intuitions.
I am a 67 superannuated person that has used PC programs for a number of years with only High School basic knowledge of trig: sines and cosines and eternity. Recently I’ve tried to explore electromagnetic science for some unknown reason. Although I have learned a great deal from your creative attempt to picture radio sinusoidal “lines”, I still cannot conceptualize how these infinite things, particulates/energy, can spread from a finite source in all directions and still fit within my brain. I can sort’v feel a singular string, an expanding and detracting coil, passing through the firmament, A string expanding in all infinite directions radio wave from an antenna doesn’t exactly fit my limited view of a sinusoidal
Can you try to give me some non-nerd picture of how a sinusoidal wave projectile is is expanded, or morphed, so it travels in all directions. Do you understand my conflict?
Very much appreciate your out-of-the-box 2 dimensional to 3 dimensional excursions. Thanx pgn
This was exactly what I needed. It was so beautiful to me that the “better graph” at the bottom in the calculus stuff I dont understand (yet) actually made me tear up. I can actually feel what I am doing now when I do trig and that helps me understand it so much more deeply. THANK YOU.
Paul Nelson: If I can butt in here, I may be able to help. Kalid can correct me if I’m wrong.
To make a conceptual switch from a two-dimensional ray to a three-dimensional emanation, think in terms of a field instead of a wave. Any radiating object, whether it’s the sun or an electron, emits a field of energy, and the field travels in all directions from the source. If you drop a rock into a pond, you get the same effect. The disturbance ripples in all directions from the point of impact.
Paul Nelson: If I can butt in here, I may be able to help. Kalid can correct me if I’m wrong.
To make a conceptual switch from a two-dimensional ray to a three-dimensional emanation, think in terms of a field instead of a wave. Any radiating object, whether it’s the sun or an electron, emits a field of energy, and the field travels in all directions from the source. If you drop a rock into a pond, you get the same effect. The disturbance ripples in all directions from the point of impact.