Also, why does nature use a unit circle?

[questionposter]

Just what the question says. What the heck about electrons and other sub-atomic particles makes them exactly correspond to a dot moving around a circle of infinite right triangles? Or sound waves even?

Edited November 8, 2011 by questionposter

[Cap'n Refsmmat]

I'm not sure how those situations imply nature uses a unit circle. Could you expand on that?

[questionposter]

I'm not sure how those situations imply nature uses a unit circle. Could you expand on that?

 

Sine waves are derived from a unit circle, nature uses sine waves.

[Cap'n Refsmmat]

You could create sine waves with the ratios of side lengths in triangles inscribed in a circle of radius 2, or 17, or 364. The ratios are the same independent of angle.

 

Using a unit circle just makes the values of sine and cosine just the lengths of the opposite and adjacent sides, respectively, because the hypotenuse is 1. When you take the ratio opposite/hypotenuse, you do opposite/1, so the sine of an angle is just the length of the opposite side of the triangle formed in the unit circle. But you make sine waves with a larger circle if you use the ratios correctly.

[questionposter]

You could create sine waves with the ratios of side lengths in triangles inscribed in a circle of radius 2, or 17, or 364. The ratios are the same independent of angle.

 

Using a unit circle just makes the values of sine and cosine just the lengths of the opposite and adjacent sides, respectively, because the hypotenuse is 1. When you take the ratio opposite/hypotenuse, you do opposite/1, so the sine of an angle is just the length of the opposite side of the triangle formed in the unit circle. But you make sine waves with a larger circle if you use the ratios correctly.

 

Yep, completely true...so how does nature use a dot moving along a circle of infinite right triangles come into play in nature? Why and how does nature an x-unit circle?

Edited November 8, 2011 by questionposter

[Cap'n Refsmmat]

I don't think nature's sitting there with a slide rule, a ruler, and a protractor when a speaker plays a sine wave into the air and makes sound waves, if that's what you're asking. We've developed mathematics because it's a tool that accurately models how many aspects of reality behave; one can't really ask "why" they behave that way, or what's "really happening" underneath, because we're limited to what we can see in experiment.

 

You might find it similarly strange that one can define the sine as:

 

[math]\sin z = \frac{e^{iz} - e^{-iz}}{2i}[/math]

 

How do right triangles relate to e -- the base of the natural logarithm -- and imaginary numbers? Does nature compute exponentials when sound waves travel through air? No.

[Schrödinger's hat]

Sine waves are derived from a unit circle, nature uses sine waves.

 

You can derive them a number of other ways as well.

I could take the odd part of [math]\frac{-i}{2}e^{ix}[/math]

I could take the taylor expansion of that as the fundamental and ask 'why does nature use infinite series?'

Or I could use a circle of different radius.

If I considered a circle of radius [math]\frac{180}{pi}[/math] and measured the ratios of the size, I'd get sin tables with the circumference of the circle being the angle in degrees.

I could even use the side lengths rather than angles. I'd have a bunch of random 360s in all my equations, but it'd still work.

I could consider a mass-spring-damper system [math]\left(\frac{d^2}{dt^2}y = -y\right)[/math] as the fundamental definition.

In this case it is something I can point to in nature (any kind of spring or r^2 potential such as gravity/electric force).

 

 

Also we only get pure sine waves when we consider special cases, usually something involving circles or some kind of springy force. Everywhere else we tend to approximate things as sine waves because they're easy.