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.
This is called smashing illusion of complexity to the grounds of simplicity.!the bestestt version .thankyu kalid:)
I’m yet thinking about acceleration and Sine!!!
It’s hard to absord that!
But I got almost everything!
Thank you!
This is really great. Unable to understand the factorial formula. But one thing I learnt is mutualfund interests are like sine waves sometimes
Hey Khalid,
I don’t get your bank account metaphor
In bank 50, income 75/week
Raise opposite your bank account
W0 50
W1 50+75-50 = 75
W2 75+75-75 = 75
W3 75+75-75 = 75
It does not seem to decrease this way
Amazing Khalid
Had you been born earlier, I am sure there would have been much less “math haters” in this world.
Keep it up
Truly speaking, same here. Don’t know why teachers usually just tell students that sine is this and that…Only the formal definitions out there. What’s the reason they don’t share such intuitive explanations in the class?