Euler's Forumla, Trig Addition, and Equating Coefficients

Click For Summary

Discussion Overview

The discussion revolves around the use of Euler's Formula to derive the addition formulas for sine and cosine through the method of equating coefficients, particularly in the context of complex numbers. Participants explore the validity and implications of this method beyond its typical application in polynomials.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant notes that while equating coefficients is typically associated with polynomials, they question whether the same principle applies when dealing with the imaginary unit i in Euler's Formula.
  • Another participant seeks clarification on what is meant by "equate coefficients" and requests details about the specific theorem being referenced.
  • A third participant asserts that there is only one addition formula for sine and mentions Euler's Formula, implying a singular approach to equating coefficients but refrains from providing a detailed explanation due to technical limitations.
  • A fourth participant explains that for complex numbers, equality requires the real parts and imaginary parts to be equal, suggesting this is the basis for equating coefficients in the context of Euler's Formula.

Areas of Agreement / Disagreement

Participants express differing levels of understanding regarding the application of equating coefficients in this context, with some seeking clarification while others assert specific points. No consensus is reached on the broader implications of this method.

Contextual Notes

Some assumptions about the nature of coefficients and their application to complex numbers remain unexamined, and the discussion does not resolve the mathematical steps involved in equating coefficients in this scenario.

elarson89
Messages
19
Reaction score
0
One can look into any precalculus book and find a proof of the addition formulas of sine and cosine. Though as most are aware there is a quick way to get the formulas by using Euler's Formula. But to get the formulas by eulers formula, you must equate coefficients with respect to the imaginary part i.

My question is this, equating coefficients was taught to be used for polynomials, because a set of coefficients uniquely determines a polynomial. How can you show the same is true with respect to i? Yes it looks very intuitive, but I'm wondering if there's something a little more powerful than that.
 
Mathematics news on Phys.org
What do you mean by "equate coefficients"? Maybe you could show us what theorem you're trying to prove to make it clearer.
 
there's, only one addition formula for sine... and then there is the euler's formula e^ix=... and only one way to equate coefficients... I would write them out but I don't know how to embed tex.
 
if you have a + bi = c + di, then by the definition of equality of complex numbers you must have a = c and b = d. so you can equate the real and imaginary parts, if that's what you're asking.
 

Similar threads

MHB Proof of Euler's Formula with No Angle Addition
  • · Replies 40 ·
2
Replies
40
Views
7K
High School Questions about equation roots
  • · Replies 17 ·
Replies
17
Views
2K
Calculating ballistics coefficient from two velocities
  • · Replies 16 ·
Replies
16
Views
4K
Undergrad Proof of double angle formulas using Euler's equation
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Quintic Polynomial Tschirnhaus Transform: Possible complex Bring Radicals
  • · Replies 0 ·
Replies
0
Views
3K
High School Simple Derivation Of Euler's Formula And Applications
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Intuitive understanding of Euler's identity?
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Pressurized containers: Stress distribution and large displacements
  • · Replies 69 ·
3
Replies
69
Views
5K
Undergrad Alternating Harmonic Numbers are cool, spread the word!
  • · Replies 4 ·
Replies
4
Views
3K
Graduate Algebraic proof that Euler angles define a proper rotation matrix
  • · Replies 6 ·
Replies
6
Views
2K