Since my engineering studies I always liked Euler’s formula, connecting sine and cosine to the unit circle in the complex number plane. That’s what Hubert’s sine-sine setting reminded me of.

I’ve been reading these for a while, I have to say I think this is the best one yet. We did Taylor series a month or so ago in my Calc class, the end of this article aided my comprehension a whole lot more than any of the class work ever did. Keep 'em coming, please!

You are a master. At some point I would love to hear your take on why the pattern that emerges in transformation matrices

cos -sin

sin cos

changes for rotations about the y axis. For now I"m feeling hungry for a salad…

“Sine waves…psssh, I know that!”

No…apparently I didn’t and you just made my life easier

@mark: Thanks – transformations matrices would be a fun addition.

@iheartcomputers: Exactly! I was the same way, I thought I understood them too :).

Kalid–great article! Dan Meyer just added a very nice modelling of a sine curve that fits with your description as sine as a smooth back and forth (or up and down). The video is here: http://vimeo.com/23798213

Hi Kalid, great article and great site! But I haven’t understand a thing: the opposite acceleration of x is the double integral of (-x). So we get that, just a moment after the beginning, sin(x) = x - x^3/3!. That’s fine.

Now we have another opposite acceleration so we have to integrate twice -(x - x^3/3!), don’t we? we then get an acceleration of -x^3/3! + x^5/5!. Summing this to the previous result of sin(x) we get sin(x) = x - 2*x^3/3! + x^5/5!. But this is obviously wrong, as the series has -1*x^3/3! instead of -2*x^3/3!.

It seems like we don’t have to integrate -(x - x^3/3!) but only x^3/3!; Why? if the acceleration must be the opposite of the current value, I expect to integrate all of its members.

Thank you very much

@simone: Great question! You might take a look at the diagrams here:

http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/

Basically, each motion (-x) creates interest (-x^3/3!), which creates s interest (+x^5/5!), which creates interest…

My intuitive understanding is that the initial motion (-x) begins a chain reaction, but the chain goes forward – it doesn’t pull back and change the original. I.e., the interest doesn’t go back and change the original… it just generates its own interest (this might be more clear in the diagrams on that article… Mr. Blue creates Mr. Green who creates Mr. Red… Mr. Blue doesn’t even know about Mr. Red!).

I struggled to teach sine to my friend, but now I realize that I didn’t understand it myself. Not intuitively anyway. Can’t wait for your next post!

@C: Thanks! Yep, often times I don’t get an intuition until visiting the topic a 2nd (or 3rd) time :).

Kalid, you are god of mathematical explanation.I have been visiting this site and I will always visit it.So many things I was taught in college is just making sense now. Thank you!

Your website is brilliant. I am 37 and have always struggled with mathematics. Reading your site makes me feel like I suffered a form of child abuse the way I was taught at school.

I left high school with the impression that sine “came from” *triangles*. University classes left me none the wiser. It was only when I bought and read books which were written fairly well that I understood sine to come from circles. It was an epiphany. I was in my mid 20s.

Now you give me another epiphany, that circles come from sine. Brilliant, but 20 years late! I wish you had been around all those years ago.

Keep it up.

Awesome!!. This changes everything!!

If there is a Chinese version, it will be better for me.

I noticed your Sine wave simulator is gone…can you find another one we can link to?

“pi is about sine returning to center! A circle is an example of a shape that repeats and returns to center every 2*pi units. But springs, vibrations, etc. return to center after pi too!”

Even Feynman never figured this out:

“About a half year later, I found another book which gave the inductance of round coils and square coils, and there were other pi’s in those formulas. I began to think about it again, and I realized that the pi did not come from the circular coils. I understand it better now; but in my heart I still don’t know where that circle is, where that pi comes from.”