Derive Sin(x+y) & Cos(x+y) Using Euler's Formula

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Discussion Overview

The discussion focuses on deriving the expansions for sin(x+y) and cos(x+y), specifically using Euler's formula. Participants express a preference for non-geometric approaches to the derivation.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant requests a derivation of sin(x+y) and cos(x+y) expansions using Euler's formula, expressing a preference for avoiding geometric methods.
  • Another participant suggests that using Euler's formula is a tedious but straightforward approach to the derivation.
  • A different approach is proposed involving 2x2 rotation matrices, indicating that this method is equivalent to using Euler's formula but operates within R^2 instead of complex numbers.
  • A participant mentions a proof available on Wikipedia, noting that it is not a complete proof but may provide useful ideas for the derivation.

Areas of Agreement / Disagreement

Participants present multiple approaches to the derivation, indicating that there is no consensus on a single method. The discussion remains open with various suggestions and preferences expressed.

Contextual Notes

Some methods mentioned may depend on specific mathematical definitions or assumptions that are not fully articulated in the discussion.

Pseudo Statistic
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Can anyone tell me how to derive the sin(x+y) and cos(x+y) expansions? The ones that are like cos x sin y or sin y cos x + other stuff?
Preferrably, could this be derived with Euler's formula alone? Or something not too geometric? (All those OAs and OBs and XBs and XYs on geometric diagrams confuse me too much to follow)
Thank you.
 
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You could use Euler's formula. It is tedious, but straightforward.
 
Another way is to use the 2X2 rotation matrices R([itex]\theta[/itex]), which have R(x)R(y)=R(x+y). This is equivalent to using Euler's formula, only you're working in R^2 instead of the complex numbers.
 
There's a short proof at wikipedia, you can view it at the end of this page.
It is, however, not a completed proof, but you can get some ideas about proving it. :)
 

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