Question 275629: Find the real part and imaginary part of (2+i)^5000
Answer by CharlesG2(834)   (Show Source):
You can put this solution on YOUR website! Find the real part and imaginary part of (2+i)^5000
2 + i convert to polar form
x + yj = r(cos θ + j sin θ)
r = absolute value or modulus of the complex number
θ = argument of complex number
2 other ways of writing the polar form of a complex number:
1. r cis θ [means r (cos θ + j sin θ)]
2. r ∠ θ [means once again, r (cos θ + j sin θ)]
r=sqrt(x^2 + y^2)
r=sqrt(2^2 + 1^2)
r=sqrt(4 + 1)
r=sqrt(5)
α = tan^(-1) (y/x)
α = tan^(-1) (1/2) = 26.57 approx
θ = 180° - α
θ = 153.43
2+i = sqrt(5)(cos153.43 + isin153.43)
2+i = re^(iθ) = sqrt(5)e^(153.43i)
(re^(iθ))^n = r^n * e^(inθ) ( by De Moivre's Formula)
r^n * e^(inθ)
sqrt(5)^5000 * e^(i * 5000 * 153.43)
2.66079147267277840928321052036*10^1747 * e^(767150i)
2.66079147267277840928321052036*10^1747(cos767150 + isin767150)
2.62036807143692290977402577219*10^1747 - (4.62041590381335873388318315839*10+1746)i is your answer
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