Vector math (small angle approximation)

In summary, the conversation discusses the approximation of sin(Θ) for small angles, with one person asking for help in understanding how to determine that Θ = Δp/p. Another person explains the concept of theta being approximately equal to the change in p over p, and a third person mentions using trigonometry to better understand the concept. They also mention the Taylor expansion of sin and how it relates to the identity function. Finally, one person suggests trying small values for theta to see the accuracy of the approximation.
  • #1
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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  • #2
Sin Φ ≈ ∅ for small angles.

(Different) Mentor edit: The above should be ##\sin(\theta) \approx \theta##, for ##\theta## in radians.
 
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  • #3
anorlunda said:
Sin Φ ≈ ∅ for small angles.
You meant to say that ##\sin\theta\approx \theta## for small angles.
 
  • #4
anorlunda said:
Sin Φ ≈ ∅ for small angles.
wait wait what? How is this true? I've never known this!
 
  • #5
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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  • #7
Boltzman 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.
 
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  • #8
Boltzman 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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  • #9
Boltzman 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.
 

What is vector math?

Vector math is a branch of mathematics that deals with the manipulation and analysis of vectors, which are mathematical quantities that have both magnitude and direction. It is used in various fields such as physics, engineering, and computer graphics.

What is the small angle approximation?

The small angle approximation is a mathematical technique used to approximate the sine, cosine, and tangent functions for small angles. It is based on the fact that for small angles, the sine and tangent functions are approximately equal to the angle itself, and the cosine function is approximately equal to 1.

How is the small angle approximation used in vector math?

In vector math, the small angle approximation is used to simplify calculations involving small angles. It allows for easier manipulation of vector components and simplifies the equations used to solve problems involving vectors.

What are the limitations of the small angle approximation?

The small angle approximation is only accurate for small angles, typically less than 15 degrees. For larger angles, the approximation can lead to significant errors in calculations. Additionally, it is only applicable to certain types of problems and may not be useful in all situations.

Can the small angle approximation be extended to other trigonometric functions?

Yes, the small angle approximation can be extended to other trigonometric functions such as secant, cosecant, and cotangent. However, the accuracy of the approximation decreases as the angle increases, so it is typically only used for small angles.

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