Intuitive Understanding of Sine Waves

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@Ram: Thanks!

@mra:

  1. Asking “How fast is sine?” is like saying “How fast is a circle?”. It’s a general shape, which you can traverse as quickly or slowly as you need. By “default” we use radians to measure angle, and get through the entire neutral-max-neutral-low-neutral cycle in 2*pi radians (6.28).

  2. “Is it really fair to say that pi doesn’t belong to circles?”. It depends on your point of view. I could imagine a world where sine was discovered first (from the motion of springs, let’s say), then pi was discovered, and later on, the shape of a circle was discovered.

  3. I think we’re saying the same thing. A pull opposite your current position vector is towards the center, which the direction of the pull. If you are at (1,0) and moving in a circular path, then your position vector is pointing East, your velocity is pointing North, and your acceleration is pointing West [towards the origin].

@Yatharth: Yep, you’ll need calculus to decipher that :). It’s basically saying “Sine accelerates your opposite of your position (if your position is x, your acceleration is -x).” To find the total distance that this negative acceleration will impact you, you integrate twice, and get -x^3/3!

Trig is basically the anatomy for circles and triangles. Learning every part of them, how they’re connected, how to find the sizes of one part given a different one.

To clarify: the variable is how fast you are along in your wave. If I write sin(pi) I mean “I am pi units along in the wave which takes 2*pi units total”, which means I’m at the halfway point. If I write sin(2x), then I am going to travel the wave twice as fast as the regular sin(x) [since I’ll be twice as far along for the same x value].

You can find sine/cosine by hand, but it’s painful. You plug in values of “x” in that infinite equation [but only take as many terms as your sanity can handle]. There are shortcuts for finding logs, sine, etc. by hand but are no longer really used, for obvious reasons. The first log tables took dozens of man-years to make.

@podAhmad: I’m not really sure what you mean by only 4 reactions or reactions to reactions. There’s an infinite sequence of them, but I only showed a few terms (with … for the continuation) in the equations.

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Here’s your chance. :slight_smile:

Note that it’s one of the much more thorough OCW Scholar courses.

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How fast is sine?
I have a question and a comment.

I get what you are saying about sin(x) going from 0 to max in pi units of time. But that is only true when you are using radians as your measure of time. And radians are defined (I think) by the angle you can get around a circle by traveling along the circumference a distance of 1 radius.

So is it really fair to say that pi doesn’t belong to circles? Without radians, there’s no pi. Without circles, there are no radians. So without circles, there’s no pi. Is that not so?

Also, you say: Circular motion can be described as “a constant pull opposite your current position, towards your horizontal and vertical center”. I don’t think that’s what you meant to say, is it? Didn’t you mean to say that circular motion is a constant pull perpendicular to current velocity? A planet orbiting on a circular path experiences a force and an acceleration that is always perpendicular to its path (and has the same strength throughout the orbit, else the orbit is an elipse).

Likewise, I think to be precise you don’t want to say sine is acceleration opposite to your curent position. Sine (one dimensional) is acceleration toward the origin, in proportion to the distance from the origin.

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good explanation!!

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This is great! Appreciate you taking so much effort to put it all together!

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@s m: thanks!

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Hi William, great question. It’s a bit circular [pardon the pun] because 2*pi is defined to be the distance around the unit circle, aka one full cycle of sine. You can “compute” the value of pi using successive approximation (see http://betterexplained.com/articles/prehistoric-calculus-discovering-pi/). Hope this helps!

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Hey Kalid, thanks for your great and inspiring posts.

I would like to have a question here. You mentioned that 2*pi is the time it takes to travel back and forth. I wonder if there is any way to deduce this? (i’m thinking of using the relation acceleration y’’=-y). Thanks so much :slight_smile:

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A few years ago I took a basic AC/DC class where I remember that we made and measured sine waves on an oscilloscope and my partner and I after we finished our project made this https://sphotos-b.xx.fbcdn.net/hphotos-ash4/292060_548653444059_1236366013_n.jpg Any chance you could tell me what we did cause we could just keep adding them on and our professor was pretty confused. It’s been bugging me for a while now since I can’t find anything similar online.

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[…] "sinusoid" is a specific back-and-forth pattern (a sine or cosine wave), and 99% of the time, it refers to motion in one […]

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Excellent explanation of the sine wave, but you seem to quickly jump through the meaning of sine in a unit circle.

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@Anon: Thanks, really appreciate it; my goal is to teach the way I wish I was taught.

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From atoms that shape spheres, to earth, to the moon, to the sea waves, sines is no other thing than the projection of the circular motion, stable, in decay or increasing, being it light, sound, radio waves, all are sinusoidal, in fact sine math as a two planes is unrealistic since most waves are tridimensional, So the circle is not a matter of geometry, geometry is one of the results of the study of circles. Men invented math not the circle, neither the waves.
If you love math, what I have always needed is an equation for the linearity of a sphere, because it would explain much of the quantum behaviour. Imagine an sphere, must be an equation that starting from any point it follows a path such that forms a perfect balance sphere. That line may cross itself but if it does you must balance the exact opposite site if there is any at all. If there is no way to do it, then my question stills. What is the balance of a particle all scientific draws as spheres, even ignoring the particles, they translate the same for atoms and talk about electron spin, and I know you know what happens when something unbalance spins. Is it he reason of the quantum behaviour?

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Hey, Kalid!

First of all, thanks for the whole site. It’s really changed my definition of “knowing” something, and it’s given me a new zest for learning!

I had a question about the pi without pictures part. I’m really interested in pi not being related to circles, but circles being related to pi. Problematically, though, I feel like I could show that sightless alien a perfectly smooth oscillation that had a period that wasn’t related to pi. I feel like a perfectly smooth shade-oscillation with period 2 seconds isn’t any less smooth than the pi version.

If the smoothness of oscillation defines pi, what makes regular sin(x) any more special than sin(x*pi) for example?

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Zeno, there is no pi not being related to circles, since pi is a constant which equals the number of times the length of the radius of ANY circle size, fits in half around the circle.
pi is the result of having found that relation between the radius and the perimeter of a circle is always the same, That is whythe whole perimeter of a circle = 2piR
You may draw a series of points in space and calculate its position from an origin based on the sine, and the final figure or equation may not result in a sine wave, then what is sine?
sine is not related to circles, but to triangles rectangles, where one angle is 90 degrees, there is a constant relation between the 3 sides, csquare = asquare + bsquare the sum of the squares of the shorter side equals the square of the longer side, no matter the size of the triangle
So the same as pi, that relation in triangles got related to angles in a 360 degrees cycle (read it cycle, not circle, this is a whole turn around something) so you can imagine something doing a whole cycle which might not be the route of a circle (radius variation if you want) so the cycle refers to 360 degrees, and sine refers to a triangle rectangle side relations, and as you can see we are not tallking neither refering to circles which in fact do not participate.

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I agree that defining sine in terms of its second derivative, or “acceleration opposite to position”, is an excellent alternative to the triangle and circle definitions, and makes the connection between sin x and $$e^x$$ much clearer.

Have you looked at the hyperbolic sine function (sinh)? It’s like a half-way house between sin x and $$e^x$$, and its graph is distinctly wave-like, though a surreal wave! Its first derivative is the hyperbolic cosine (cosh), but its second derivative is itself. Acceleration is equal to position, as in the case of $$e^x$$.

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Hi Stephen, great comment. To be honest, my intuition for sine/cosine isn’t as strong as the ones for e, Pythagorean Theorem, etc. so I’m a bit in the “discovering” vs “deriving” mode myself.

I might say something like this: suppose we have a concept of perfectly smooth growth, which is epitomized by e^x.

If we combine the idea of perfectly smooth growth (e^x) with rotations (imaginary numbers), we get e^ix.

Intuitively, before graphing anything, we should imagine that e^ix results in something along the lines of “perfectly smooth rotation”.

What would this shape be? Well, it should be symmetrical (why would it favor one side over the other?). It should embody the essence of rotation, spinning. And so on.

Pretty soon, we might see that a circle is the shape which satisfies this intuition. Now the question of sine/cosine comes in.

This circle exists in 2d: if we analyze each dimension independently, it seems like each dimension should be moving perfectly smoothly as well (again, why would one be favored over the other?). We can’t rotate in a single dimension, but whatever motion we have, should be smooth.

That pattern of motion, the smooth sway in a single dimension, can be called sine. And we can work out that

e^ix [perfectly smooth rotation] = cos(x) [smooth sway in the horizontal direction] + i*sin(x) [smooth sway in the vertical direction]

We’ve separated our 2d motion into a combination of two 1d trajectories. Getting into even more nitty gritty, the series expansion of e^ix = series expansion of cos(x) + series expansion of i*sin(x)

That is… cosine and sine can literally be “factored” out of the combined circular path we see in e^ix.

Hopefully that helps?

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Wow…My mind is blown.No one ever teaches students these things,and they can make a world of a difference.

Your articles are phenomenal and so is your attitude.

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thanks it helped me alot :slight_smile:

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better explained!!!