MHB Complex Number Problems in Applied Maths

[anil86]

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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[MarkFL]

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]

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.

 

[Opalg]

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. $x^7 + x^4 + x^3 + 1 = x^4(x^3 +1) + x^3+1 = \ldots$.

 

[anil86]

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