Thanks Colin! Great point with the hyberbolic versions – I’ve only dabbled in them very briefly. I like that they can be defined in terms of exponentials… they’d be a fun topic to get into :).
Thanks, you rock
@Stephen: Thanks! Great question on the connection between sine and triangles. I actually prefer starting with circles, and then seeing how triangles fit in.
Let’s say we’re traveling around the unit circle (radius 1). As we go around, we have some height above the x-axis. From this discussion, we can see that a circle is made up of two “sways”, one controlling the vertical position (call that sway sine), and another controlling the horizontal position (call that sway cosine).
The only difference between the sways is where they start: sine starts neutral, and starts moving up. Cosine starts at its max value, and starts moving towards center. For now, let’s think about sine, our vertical position.
If we take our spot on the circle and extend a line down, and to the center of the circle, we end up making a right triangle! The height, our sine, is the side “opposite” to the angle, and the line to the center is the hypotenuse. The resulting triangle must be a right triangle, because we drop a line straight down from our current position.
Now, by an accident of history, we started working with triangles before circles, so we found this “vertical pattern” (sine) in triangles, then built up to circles. But again, I’d prefer to start with circles and work down to the triangles buried inside.
Another way to put it: the swaying vertical motion of a circle, called sine, can also be seen in the swaying heights of the right triangles formed along our circular path.
(The last step is to allow for any hypotenuse, not just the unit circle, so we scale by “H” to show that sine is present in any circle no matter how large, similar to how pi is present in any circle, no matter the radius.)
…in the same way the you very intuitively explained the Taylor series for e. In that case, I could visualize what each term corresponded to.
Hi Kalid,
Thanks again for some more great insights. Very helpful and definitely adding to my understanding of sin.
So I think I see everything up to the point where we make the leap from rectangular values (x’s and y’s) to polar values (theta). I think the gap in my intuitive understanding of sin stems from not being able to see where the Taylor series of sin comes from.
I think your article is great at teasing out the corroborative intuition on the Taylor series: given the existence of the series itself, you demonstrate that the alternating positive and negative terms are corrective tugs in opposite directions, which gives rise to the wave appearance. I get that, and I think it’s a great insight.
But do you think there is a way to draw a diagram with some triangles or circles or something, so we can start with rectangular x’s and y’s, and derive the Taylor series intuitively, which we THEN define to be the function called “sin(x)”? (rather than say the commonly used circular derivation, where the definition of sin and cos is a given, and based on the fact that each function’s derivative is the negative of the other, you come up with a Taylor series to satisfy that condition. That’s a corroborative approach to be)
I like how you point out that something more ‘fundamental’ is going on with the trig functions that goes beyond circles: periodicity. But at the end of the day, are the Taylor series of these functions capable of being derived without using the visualization of a circle?
The person who asked this question on math.stackexchange is getting at what I’m looking for:
However, the most up-voted explanation here is, to me, circular and un-intuitive. It employs the definition of sin and cos and some trig identities to show where the Taylor series came from, which is to me unhelpful.
Let me know if I’m chasing ghosts. Thanks again for your time, Kalid
Best,
Stephen
Hi Kalid,
Thank you so much for your speedy reply. I really appreciate it.
I really see what you mean when it comes to thinking of sin and cos as ‘swaying’. I think I’m starting to really internalize that way of looking at it, which definitely is helping to develop my intuitive grasp of these functions.
However, I think my intuition is still at the stage where I begin with the assumption that circles are made up of a perfect balance of two complementary sways as described by sin and cos, and THEN work backwards to intuitively confirm this initial assumption.
Ideally, I would like my intuition to see how sin and cos connect to circles without first assuming that they do, and then finding different ways to confirm that this is in fact the case.
For example, your article on e delved into the math in an incredibly intuitive way, to the point where I saw exactly how the general concept of growth fell directly out of the details of the equations. Or with your article on Pythagorean theorem, where you explicitly highlighted the physical areas that were implicitly being added, which accounted for the squared terms in the equation.
So I’m wondering if you think it’s possible to approach sin and cos the same way, where circles can kind of ‘fall out’ of the math. Because right now I’m at the stage where my intuition is only confirming, rather than deriving. Perhaps if you know any of the history of how these functions were first discovered…that is often very helpful in developing intuition that derives rather than confirms.
Please let me know if this isn’t making sense and I will try and elaborate more. I’m being a little bit vague because I’m still trying decide if I’m at the point where I need to accept something as empirically true, or if I can reduce my understanding to more fundamental principles.
Thanks so much for bearing with me, Kalid. I appreciate your work so much
Best,
Stephen
Firstly, I’d like to thank you for putting in the time & the work to even be able to explain math in this way. It helped me very much in a time of need. 
Secondly, I apologize for not going through all your comments to see if this question had already been answered, but could you possibly explain tangent, secant and cosecant wave patterns on a graph? Or if you’ve already answered that somewhere, post the link to the page.
Much appreciated.
Cheers.
@Stephen: Whoops, I lost track of this comment, I’d like to cover it, probably in a follow-up article 
@Monq: I don’t have anything on the graphs, but http://betterexplained.com/articles/intuitive-trigonometry/ covers the meaning of the various functions. I’d like to do a follow-up on the graphs as well.
[…] yesterday, we start off with the energy of a sine wave, coming out of the deep west of my lady’s field, stretching across to the center of may. […]
[…] yesterday, we start off with the energy of a sine wave, coming out of the deep west of my lady’s field, stretching across to the center of may. The […]
[…] yesterday, we start off with the energy of a sine wave, coming out of the deep west of my lady’s field, stretching across to the center of may. The […]
Your explanation of how SINE originated is brilliant. My husband (has PhD) and I have Master’s degrees and never thought of this explanation until our highschool daughter asked me this. I am very glad that you have this site.
Thanks GV, really glad you enjoyed it! I didn’t start seeing the meaning of Sine until maybe a decade after learning it “officially”.
[…] “正弦曲线”意味一种往复运动模式（正弦或余弦函数）。在99%的场合里，指的都是发生在象限里的运动 […]
[…] “正弦曲线”意味一种往复运动模式（正弦或余弦函数）。在99%的场合里，指的都是发生在象限里的运动 […]
You are a hero for sharing this explanation. Thank you so much.
You might like my Sin wave visualization (about half way down on this page). Please feel free to do whatever you want with it (and if you make it better, let me know!)
About 10 years of studding (from the first day sine was introduced to us in high-school) and ‘today’ I actually learned what sine really is. Dear kalid, you are the best teacher ever. I wish others were like you too.
Seriously this math better explained… Thank you for the time and efforts
Excellent
please explain pi in Terms of fundemental concepts