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Trigonometry The Unit Circle, the Sinusoidal Curve, and the Slinky...

m3dicat3d

New member
Jan 11, 2013
19
I seem to recall when taking college Trigonometry my professor saying that the unit circle and sinusoidal curves were basically a mathematical represention of a slinky in that the unit circle was the view of a slinky head on, so that what you saw in the two dimensional sense was a circle, and that when you looked at the slinky from the side (non-compressed of course) you saw the sinusoidal curve.

It's been a while, but I think he mentioned this to help us make the connection (after we had learned the unit circle) to then graphing the sine and cosine fucntions, and how the graphs/values came directly from the unit circle itself (in other words, we were looking at a different side of the same coin so to speak).

I ask because I am helping a student right now in his trig class, and now that he himself has mastered the unit circle, he is moving into graphing the trig functions (sine and cosine at this point) and I'm looking for any little thing that will help him make these connections as well.

I just don't want to use the slinky analogy if that is in fact not true, so I thought I'd defer to those here who clearly know their math better than I. Thanks :confused:

Edit: Here's something I was wanting to show my student in this regard and serves to illustrate what I'm saying

Daylight Map circular.JPG

Daylight Map Sine wave.JPG
 
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Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,492
I am not sure right now if a slinky viewed from the side is a sinusoidal curve. This requires taking into account some physical considerations regarding how exactly a slinky stretches. I am also not sure about the solar shadow on the Earth because taking the curvature of the Earth into account may be tricky (or it may be not). But you can find animations that illustrate the actual definitions of sine and cosine. See the Wikipedia pages and especially the following GIF:



Here the horizontal axis represents the angle, which increases linearly, and the vertical axis represents the vertical coordinate of the point rotated around the origin.

The only thing that I don't like about GIFs is that, as far as I know, one cannot control the speed of animation. One can stop the animation by pressing ESC, but restarting it apparently requires reloading the page.
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,774
Welcome to MHB, m3dicat3d! :)

Same here, I'm not sure if the slinky from its side is really a sine (there's some material science involved there), but I am sure that it can easily be modeled as a sine.

As for the side view of the earth, even though the shadow edge forms a circle, the projection of the earth on a flat surface is a difficult one.
The result will not be a sine.
It only looks a bit like a sine.

Another educational value of the slinky is that the unit circle extends beyond 2pi.
It's like entering a new loop of slinky.
Although like this you should use a compressed slinky.

Btw, the gif automatically keeps repeating for me.
 
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m3dicat3d

New member
Jan 11, 2013
19
Thanks everyone, both thread replies and pm's...

hehe, I meant this strictly as an analagous scenario, as getting into materials science or the fact that the earth is not perfectly spherical would likely complicate matters for my student.

The other reason I've asked this though, is for my own edification as this comparison from my professor has always made me picture (bringing this over into the realm of physics) a wave being more of a helix, than simply a 2-D wave representation in something like a textbook example figure.

I never did follow up on this (especially at the time as trig was so new to me I was only trying to keep up :D ) but as this opportunity with my student has presented itself, it makse me wonder again.

I can think of an example of a charged particle (a proton for instance) moving along a parallel magnetic field with a motion that "circles" the field line as the particle travels in the direction of the field line. If viewed from a bird's eye view, the proton would trace out a "sine-like" trail/curve (emphasis on "like" - I'm keeping this as simple as possible without invoking the nuances that would affect amplitude and frequency, etc.). But if you viewed that same scenario dead-on, looking straight down the filed line (like a cross-product going into the page) you would see the proton simply moving in a repeating circle.

Here's a link I just dug up in case it helps:

Charged particle in a magnetic field - YouTube

I hope this makes sense? This is now for my own understanding lol. I wouldn't want to bring up ANYTHING that would complicate the matter for my student, so to repeat, now the question is more self-serving :D

I've just always wondered with all the math and science I had for my degree about some of these connections (if indeed these are actually connected) that were never really brought up to me as a student at that time.
 
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Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,774
It's not so much that the earth is not a perfect sphere (it's close enough) - but even a perfect sphere can't be mapped "perfectly" on a plane.
The result really is not a sine.
Just draw a real sine over it and you'll see.

And yes, in physics these things really behave like the slinky - we only "see" the sine from the side, but there is also a cosine involved.
An electromagnetic wave (read: light) has an electric and a magnetic component that switch roles just like a sine and a cosine.
Periodic electrical phenomenons are typically modeled as a wave with a real and an imaginary component.
Similarly a pendulum switches its energy back and forth between potential and kinetic energy just like the combination of a sine and a cosine.
 

m3dicat3d

New member
Jan 11, 2013
19
Thanks!!!

I've always wondered about this! Even years after my degree lol.

Yeah, with mapping projections that either highlight areal projections with more accuracy or geographic borders that reflect the shapes more closley, I see where you're coming from about projecting a sphere onto a plane (like the limitations a GIS would have with attempting to project spherical regions accurately depending on the coordinate projection system you're using). This is more for just a vanilla analogue like I said. But hopefully it will still be useful to my (or perhaps other students) to see a real world application of what they are working with, to help them retain or even make the connections between the UC and the Sine/Cosine curves.

Anyhow, I'm blathering now, but THANKS again so much, I learned something new today!!!
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,774
I've always wondered about this! Even years after my degree lol.
That's nothing new to me.
I'm many years beyond my degree as well and I keep learning new stuff on forums like this one.
And that is about stuff I thought I already knew quite well too!
 

m3dicat3d

New member
Jan 11, 2013
19
One more thing about the video link above...

Is it me, or did the animator "forget" (so to speak) about right hand rule? I saw a couple of other comments about this at you tube, but no responses? Maybe there is something I'm forgetting about?
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,774
One more thing about the video link above...

Is it me, or did the animator "forget" (so to speak) about right hand rule? I saw a couple of other comments about this at you tube, but no responses? Maybe there is something I'm forgetting about?
No, the animator has the right direction.

The Lorentz force is $F = q ( \mathbf v \times \mathbf B)$.
That is, if you rotate the velocity onto the direction of the magnetic field with the fingers of your right hand, then your thumb gives the direction of the Lorentz force.
This direction is indeed to the center of the circular path.

Now if the charge was making that path in the absence of a magnetic field, it would induce a magnetic field in the opposite direction.
 

m3dicat3d

New member
Jan 11, 2013
19
thanks again :D