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Complex Numbers

anil86

New member
Nov 1, 2013
10
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
Image0319.jpgImage0319.jpg
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
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?
 

anil86

New member
Nov 1, 2013
10
Solve equations using De Moivre's theorem:

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

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


I tried multiplying with (x - 1). Also tried putting x^3 = y; didn't work.
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,703
Solve equations using De Moivre's theorem:

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

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


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

anil86

New member
Nov 1, 2013
10
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