Trig functions and the gyroscope

[Trying2Learn]

TL;DR Summary

Why do trig functions cancel?

Good Morning

As I continue to study the gyroscope with Tait-Bryan angles or Euler angles, and work out relationships to develop steady precession, I notice that the trig functions cancel.

I stumble on terms like:

1. sin(theta)cos(theta) - cos(theta)sin(theta)
2. Cos_squared + sin_squared.

Can anyone tell me why these appear so often?

Yes, I see it happening in the math; I see it happening so often that when I stumble on a term like
Cos_squared MINUS sin_squared. Then, I say to myself: "I must have made a mistake."

Why do things like this happen? Why is it "to be expected" that in such analyses, the terms should simplify? It cannot be serendipity - something must be happening.

ADDITION TO POST: I can sort of reason out the second one (cos_squared + sine_squared); I sort of reason that machines operate in circular motion and that equation is a re-definition of a circle of unit radius (so it is expected to appear). But I cannot rationalize why the first equation appears so often, canceling for simplicity of the expressions.

 

[Baluncore]

Look at it with a sense of humour.
There are two alternate paths. One to the left, another to the right. They meet behind you, where they cancel.

https://en.wikipedia.org/wiki/Humour
The benign-violation theory, endorsed by Peter McGraw, attempts to explain humour's existence. The theory says "humour only occurs when something seems wrong, unsettling, or threatening, but simultaneously seems okay, acceptable or safe".

 

[Filip Larsen]

As I continue to study the gyroscope with Tait-Bryan angles or Euler angles... and work out relationships to develop steady precession, the trig functions cancel.

Could you provide an example of what you refer to? It is quite normal and desired to use mathematical relations to simplify expressions as much as possible after analysis, trigonometric expressions included.

 

[Trying2Learn]

Could you provide an example of what you refer to? It is quite normal and desired to use mathematical relations to simplify expressions as much as possible after analysis, trigonometric expressions included.

Well, I do not know how to use the equation editor here, so I am limited. The best I can say is that if I go down the path of Euler angles (say, rotation about one axis, then a second, then a third) and mutiply them and their rates, I obtain huge equations. However, closer inspection of these equations shows certain patters, and buried in them, are these equatios, and they start canceling, leading to much simpler expressions.

I attach a picture of one sequence of the math I work out, leading to the gyroscopic effect. Whenever I work this out, I see such simplifications, happening (so much so, that when they do not happen, I know to go back and find my error)

But just to emphasize: I am NOT asking someone to check my work. The work is fine. I am asking a question that is a bit more philosophical. Why should I expect this to happen, and NOT consider it to be good luck?

 

[Filip Larsen]

However, closer inspection of these equations shows certain patters, and buried in them, are these equatios, and they start canceling, leading to much simpler expressions.

Well, I think you have already answered your own question by originally observing that it is identities and relations that are used to simplify expressions. So, when you ask why, say, some trigonometric expressions cancels or simplifies it is indeed because of those identities. So the question then becomes "why do these trigonometric identities exist" and for that one probably have to work through their derivation to see why in a mathematical sense.

Note that often big complicated expressions only simplify in an approximate form. For instance, for $\theta\approx 0$ we can make the first order approximation $sin\theta\approx \theta$.

 

[Trying2Learn]

Well, I think you have already answered your own question by originally observing that it is identities and relations that are used to simplify expressions. So, when you ask why, say, some trigonometric expressions cancels or simplifies it is indeed because of those identities. So the question then becomes "why do these trigonometric identities exist" and for that one probably have to work through their derivation to see why in a mathematical sense.

Note that often big complicated expressions only simplify in an approximate form. For instance, for $theta\approx 0$ we can make the first order approximation $sin\theta\approx \theta$.

Thank you (seriously, thank, you, but)

You wrote: "So the question then becomes "why do these trigonometric identities exist?"

But I am asking: "why do these trigonometric identities appear in the analysis of the gyroscope?"

I can reason out one of them, I THINK (cos_squared+sin_squared as a circle), but when the other appars with a PLUS sign I know I made a mistake. But I cannot figure out why I know (or expect) this.

 

[Filip Larsen]

why do these trigonometric identities appear in the analysis of the gyroscope?

OK, that is a more narrow question that someone here might have an interesting answer to.

I am probably only able to provide the "meta-answer" that complex symbolic analysis that can be simplified often are far more well-known or used than symbolic analysis that does not offer any particular simplification even if the latter would be more accurate, so it is not a surprise that the analysis you find in text books ends up with some suitably simple expressions. Or in other words, you don't often see analysis that ends up in big and ugly expressions because it is not very useful for most purposes.

 

[Trying2Learn]

OK, that is a more narrow question that someone here might have an interesting answer to.

I am probably only able to provide the "meta-answer" that complex symbolic analysis that can be simplified often are far more well-known or used than symbolic analysis that does not offer any particular simplification even if the latter would be more accurate, so it is not a surprise that the analysis you find in text books ends up with some suitably simple expressions. Or in other words, you don't often see analysis that ends up in big and ugly expressions because it is not very useful for most purposes.

OK, so someone has sawed off my left arm and it hurt. And I have a hangnail on the thumb of my right hand.

Your comment soothes my missing left arm; but this hangnail on my right hand is now really irritating me

I am almost happy.