Proof of trigonometric multiplication of complex numbers

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SUMMARY

The discussion centers on the proof of trigonometric multiplication of complex numbers, specifically focusing on the multiplication process and its implications. The user successfully derived the expression cos(alpha-beta) + i*sin(alpha-beta) but struggled to recall the fundamental identity cos² + sin² = 1, which is crucial for simplifying the results. This highlights the importance of foundational trigonometric identities in understanding complex number operations.

PREREQUISITES
  • Understanding of complex numbers and their representation in polar form
  • Familiarity with trigonometric identities, particularly cos² + sin² = 1
  • Knowledge of Euler's formula relating complex exponentials to trigonometric functions
  • Basic skills in algebraic manipulation of trigonometric expressions
NEXT STEPS
  • Study the derivation of Euler's formula: e^(iθ) = cos(θ) + i*sin(θ)
  • Explore the geometric interpretation of complex multiplication
  • Learn about the applications of trigonometric identities in complex analysis
  • Investigate the relationship between complex numbers and polar coordinates
USEFUL FOR

Students of mathematics, particularly those studying complex analysis, trigonometry, and algebra, as well as educators looking to clarify the multiplication of complex numbers using trigonometric identities.

embassyhill
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This is supposed to be a proof of trigonometric multiplication of complex numbers:
0I9sN.png

What happened at the =...= point? I understand everything up to that.
 
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embassyhill said:
This is supposed to be a proof of trigonometric multiplication of complex numbers:
0I9sN.png

What happened at the =...= point? I understand everything up to that.

Do the multiplication and see what happens. Bottom part is just equal to r.
 
I did manage to get cos(alpha-beta)+isin(alpha-beta) on the upper part but I was too dumb to remember cos^2+sin^2=1 :P. Thanks.
 

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