How can the Euler formula be used to derive the properties of rotating vectors?

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    Rotating Vectors
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Discussion Overview

The discussion centers on the derivation of properties related to rotating vectors using the Euler formula. Participants explore the trigonometric identities involved and seek clarification on their proofs, particularly in the context of how these identities relate to the Euler formula.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant requests clarification on the derivation of specific trigonometric identities related to rotating vectors.
  • Another participant asserts that the equations in question are basic trigonometric identities and points out an error in the cosine expression, suggesting a correction is needed.
  • A later reply acknowledges the correction and expresses a desire to understand the proof of the identities, indicating an interest in the underlying principles.
  • One participant suggests that familiarity with the Euler formula allows for a quick derivation of the identities using the sum of angles.

Areas of Agreement / Disagreement

Participants generally agree on the nature of the equations as trigonometric identities, but there is a disagreement regarding the correctness of the original cosine expression, which one participant claims contains an error.

Contextual Notes

The discussion does not resolve the proof of the identities or the implications of the Euler formula, leaving these aspects open for further exploration.

Fjolvar
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I copied a diagram from my book of rotating vectors, and I just want to know how they got the following:

A cos(theta-phi) = A(cos(theta)cos(phi)+sin(theta)sin(phi))

and

A sin(theta-phi) = A(sin(theta)cos(phi)-cos(theta)sin(phi))

Which properties were used?

Thanks.
 

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Last edited:
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These are basic equations, that hold for sines and cosines in general, which are taught in any trigonometry course.

Note: there is an error in your cosine expression. One of the - signs should be +. You have to change one sign, but not both.
 
mathman said:
These are basic equations, that hold for sines and cosines in general, which are taught in any trigonometry course.

Note: there is an error in your cosine expression. One of the - signs should be +. You have to change one sign, but not both.

Thanks, I meant to write a + sign. So I guess I'll just understand this as an identity. I'm still curious how it's proven.
 
Last edited:
If you are familiar with the Euler formula [eix = cosx + isinx] you can derive it very quickly using x = a + b.
 

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