Vector math (small angle approximation)

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Discussion Overview

The discussion revolves around the small angle approximation in vector mathematics, particularly focusing on the relationship between angles and changes in momentum, represented as Θ = Δp/p. Participants explore the implications of this approximation and its mathematical foundations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant seeks clarification on how to derive Θ = Δp/p from the equation p + Δp = p', expressing confusion about the relationship.
  • Several participants mention the small angle approximation, noting that for small angles, sin(θ) can be approximated as θ.
  • There is a discussion about the Taylor expansion of sin(θ), with one participant explaining that for small values of θ, the higher-order terms become negligible.
  • Another participant emphasizes the importance of using radian measure for angles when discussing the approximation.
  • Participants suggest testing the approximation with small numerical values of θ to observe its accuracy.
  • Some participants express uncertainty about the small angle approximation and its validity, questioning its general acceptance.

Areas of Agreement / Disagreement

While there is some agreement on the validity of the small angle approximation, uncertainty remains regarding its application to the specific problem of Θ = Δp/p. Multiple viewpoints are presented without a clear consensus on the derivation or implications.

Contextual Notes

Participants reference the Taylor expansion and derivatives of the sine function, indicating a reliance on mathematical definitions and approximations that may not be universally accepted or understood in the same way.

Boltzman Oscillation
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Given the following vectors:

vectors.png

how can i determine that Θ = Δp/p ?
I can understand that p + Δp = p' but nothing arrives from this. Any help is welcome!
 

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Sin Φ ≈ ∅ for small angles.

(Different) Mentor edit: The above should be ##\sin(\theta) \approx \theta##, for ##\theta## in radians.
 
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anorlunda said:
Sin Φ ≈ ∅ for small angles.
You meant to say that ##\sin\theta\approx \theta## for small angles.
 
anorlunda said:
Sin Φ ≈ ∅ for small angles.
wait wait what? How is this true? I've never known this!
 
Delta2 said:
You meant to say that ##\sin\theta\approx \theta## for small angles.
Ah I guess I could see that being true since the taylor expansion of sin is theta - theta^3/3! +theta^5/5! so a small theta would cause the terms after the first to be significantly small.
 
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To be clear, the angle is in radian measure.
 
Boltzmann Oscillation said:
Ah I guess I could see that being true since the taylor expansion of sin is theta - theta^3/3! +theta^5/5! so a small theta would cause the terms after the first to be significantly small.
You also see that from the fact that sin(0) = 0, sin'(0) = 1 and sin''(0) = 0. So sin is equivalent to the identity function up to the 2nd derivative around 0.
 
Last edited:
Boltzmann Oscillation said:
Ah I guess I could see that being true since the taylor expansion of sin is theta - theta^3/3! +theta^5/5! so a small theta would cause the terms after the first to be significantly small.

It's instructive to take ##\sin(\theta)## for some small values of ##\theta## (always in radians) to see just how good the approximation is. Even at ##\theta = 0.1##, which is a little larger than what we usually consider "small compared to 1", it's a pretty good approximation.
 
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Boltzmann Oscillation said:
how can i determine that Θ = Δp/p ?
Are you looking for more than just basic Trigonometry here?
 
  • #10
sophiecentaur said:
Are you looking for more than just basic Trigonometry here?
Well the other gentlemen helped me understand better now but maybe you can provide more insight. So, yes.
 
  • #11
RPinPA said:
It's instructive to take ##\sin(\theta)## for some small values of ##\theta## (always in radians) to see just how good the approximation is. Even at ##\theta = 0.1##, which is a little larger than what we usually consider "small compared to 1", it's a pretty good approximation.

okay i will try some small numbers. Thank you sir.
 

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