@Theo: Wow, thanks for the kind words! I’m basically doing the same as you – going back to relearn what I thought I learned in college :).

@Michael: Thanks for the kind words and encouragement! There are so many misconceptions that I’m only beginning to unravel (this whole sine business only started clearing up in the last few months). I love that epiphany feeling. Sorry for the delay in reply, I was on vacation when these earlier comments were posted.

@Jay: You’ll never look at sine the same again :).

@Zack: Don’t know Chinese unfortunately… Google translate?

You are awesome!!! Seeing sine as motion and not part of a static geometric diagram is so new! I’ve been really curious about how to understand math in different ways. I see at uni that the concepts learned in school are gussied up in different disguises, based on the discipline.

I just had a question to clarify what you said about pi as being a notion of time. I didn’t quite understand, given that I’ve always thought of pi as distance. But this is sort of a two-sides of a coin thing…do you mean that distance=rate x time, and just assume that the unit for the rate of sine to go from neutral to neutral is pi/seconds? So that the units cancel out and thus distance is equal to time?

Also, could you just briefly clarify how you got the successive terms in “The Better Models of Sine” section? x is the initial impulse, and I think I follow your double integration of the opposite position, but then where do you get x5/5! and so forth?

@Ashley: Thanks for the note! Yep, a key to math is trying to see it from different angles – some click better than others.

Yep, pi is like time if you assume rate = “1 unit per second” (or really, 1 unit per unit time). So then you have distance = rate * time, or distance = time.

If we’re talking about radians (distance traveled along the outside of the circle) then if we go for 2pi units of time, we’ve traveled a distance of 2pi and we’re all the way around. If you see sine as a “living, moving” process, the 2*pi is how often the process is back to its initial position. pi by itself is the time from neutral to neutral (middle-top-middle or middle-bottom-middle).

Great question. With e, we’ve seen that our “interest earns interest”. In the same way, the “restoring force” of sine creates its own restoring force, which creates another restoring force… and so on.

The thing is, these happen simultaneously, we don’t need to wait for the first force to happen (or maybe another way, it happens infinitely quickly so we can’t tell). So when we compute sine, we have to account for as many restoring forces as possible to be accurate.

So, our original force is x (call this A). This creates a restoring force B (pulling us to center) of x^3/3! (the double integral of x). That creates force C which tries to balance the pull center with a push away (double integral of x^3/3! is x^5/5!).

In this way, A creates B, which creates C, which creates D… to infinity :). Actually, each restoring force is in the opposite direction (x has restoring force -x^3/3! because the rule for sine is your acceleration, your second derivative, is the opposite your current position).

I was wondering if you might be able to offer any intuition as to why sine of an angle is also the ratio of the opposite side to the hypotenuse in a right triangle? I get that right triangles would have consistent ratios for the different side lengths, but how is sine (and cosine, tangent) able to describe these ratios?

Sine is the point that rotates around another fixed point which causes motion in a linear fashion on a line bisecting the fixed point? Like the arm affixed to the wheels of a steam locomotive running along a track?

I don’t get it still… Worse yet is how, from this, am I to make heads or tails about trig identities? They seem to be everywhere…

Woah! This is awesome! I can’t believe I found this website yesterday and I was like “Too bad I didn’t find this website some time before my exam, would’ve made my preparations much more fun and versatile”. But then I go to the exam, and it turns out I got the wrong date- it’s actually tomorrow. So I have this whole day free to read a bunch of your articles. They make me all excited about math again! Yeepeee!
Thank you sir, you are a blessing!

So I was seriously taking on trig (apparently it’s a prerequisite for calculus), and I came upon trig and family. 3 sides of a triangle having 6 relationships called Sine, Cosine, Tangent, Cotangent, Secant, Cosecant. Is that it? Is that all there is to trig apart from learning to manipulate them?

I read your article a couple of times, but I still didn’t get a few things. You said the x in sin(x) was how far along it was in a wave. But then you said the x meant how fast in relation to the ‘normal’ wave which took π seconds to return to neutral. What gives?

And why can’t we calculate all of these by hand? I mean, sine and cosine and all are functions, right? They must have an exact definition. Why can’t we figure them out by hand? Why do we need these ‘log books’? That reminds me. Is there a better way to find logs than tables? Some sort of a approximation method maybe…

Correction: sine is bounded between -1 and 1, inclusively, if the argument is a real number. In fact, there are (infinitely many) complex solutions for sin(x) = 2. For example,
sin(π/2 + i*ln(2 + √3)) = 2

@Anonymous: Great question. I see sine as a general sway back and forth. We can notice this sway on a circle by realizing the height of a point sways up and down if we just look at that axis. It’s a bit like the arm on the wheels on a locomotive, yep. (But again, that’s just an example – sine is a general concept which shows up everywhere! Some of the confusion is around the definition of sine (it’s a sway!) vs. examples (it shows up in circles, and pendulums, and so on).

Trig identities are another beast entirely. Basically, it turns out that one “sway” (sine) can have relationships to other sways (cosine), for example sin^2 + cos^2 = 1. I’d like to cover this more!

@Christian: Although it is true that sine takes on values outside of [-1,1] for complex arguments - and in fact takes on all complex values (more generally, this is true, with the exception of at most 1 value, for any non-constant analytic complex functions, by Picard’s theorem http://en.wikipedia.org/wiki/Picard_theorem), I would argue that this critique is misplaced here as it refers to a mathematical extension of the definition of sine to the complex plane, a space of consideration that was never once intended to be mentioned here. Sine was originally constructed to carry a certain meaning - to represent the path of harmonic motion (modulo amplitude) (which, by definition, doesn’t “leave” its interval of oscillation) and parameterize the unit circle (which, again by definition, cannot have a distance from the center greater than 1).

This lesson aimed to explain the origin of, intuition behind, and exposition of: the sine wave, not the general rigorous properties of the sine function over the complex field (which belongs in a class on complex analysis, not in a blog post on high school algebra).

In addition, @Kalid only mentions these “bounds” (he, in fact, never uses this more formal term that carried the very baggage that you intended to dispose of) within the language of “moving” and “swaying” - these already imply that we are on the real line: intuitively this is because motion implies time (or at the very least - 1 dimensional travel) which, for all intents and purposes, is “real” (as C is 2 dimensional - we cannot “trace” across the entire vector field of sine over C).

Often concepts originally defined to mean something spacial/physical get generalized with mathematical language, and along the way lose some of the properties that originally motivated their definition in the first place - this does not mean that the explanation (or even conception) of them in those original terms is “incorrect” - there is no “one true sine function” but rather a general notion of sine with distinct appropriate definitions for different contexts. So, correction: your comment was just an interesting addendum.

g88888888888888 post as usual specially the series explanation is awesome.
But here comes a question!!!
From 0 to pi there are only four reactions or reactions to reactions…but shouldnt it be a continuous chain of reaction, i mean there should be infinite terms between 0 to pi.