MHB Complex Number Problems in Applied Maths

AI Thread Summary
The discussion focuses on solving complex number problems in applied mathematics, specifically proving identities involving cosines and using De Moivre's theorem for polynomial equations. Participants are encouraged to share their attempts to solve the problems, which include equations like x^7 + x^4 + x^3 + 1 = 0 and x^7 - x^4 + x^3 - 1 = 0. Factorization techniques are highlighted as effective methods for tackling these equations. One user successfully solved the problems by factorizing and applying De Moivre's theorem. The thread emphasizes collaborative problem-solving in the context of complex numbers.
anil86
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This is a thread for complex number problems in applied mathematics.

1. Prove that: 1 + cos x + cos 2x + ...cos (n - 1)x
= {1 - cos x + cos (n - 1)x - cos nx} / 2 (1 - cos x)

= 1/2 + [{sin (n - 1/2)x}/2sin (x/2)]2. If a = cos x + i sin x, b = cos y + i sin y, c = cos z + i sin z, prove that

{(b + c) (c + a) (a + b)}/abc = 8 cos (x - y)/2 cos (y - z)/2 cos (z - x)/2
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I have moved this topic here to our Pre-Calculus sub-forum as it is a better fit than Number Theory.

Can you show what you have tried so our helpers know where you are stuck and what mistake(s) you may be making?
 
Solve equations using De Moivre's theorem:

1. x^7 + x^4 + x^3 + 1 = 0

2. x^7 - x^4 + x^3 - 1 = 0I tried multiplying with (x - 1). Also tried putting x^3 = y; didn't work.
 
anil86 said:
Solve equations using De Moivre's theorem:

1. x^7 + x^4 + x^3 + 1 = 0

2. x^7 - x^4 + x^3 - 1 = 0I tried multiplying with (x - 1). Also tried putting x^3 = y; didn't work.
Those polynomials factorise, e.g. $x^7 + x^4 + x^3 + 1 = x^4(x^3 +1) + x^3+1 = \ldots$.
 
Opalg said:
Those polynomials factorise, e.g. $x^7 + x^4 + x^3 + 1 = x^4(x^3 +1) + x^3+1 = \ldots$.

Hi Opalg,

I solved it by first factorizing & then using De-Moivre theorem as suggested by you. Thank you.

Anil
 
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