Question

According to De Moivre’s theorem what is the value of z^1/n ?

(a) r^1/n [cos(2kπ + θ) + i sin(2kπ + θ)]

(b) r^1/n [cos(2kπ + θ)/n – i sin(2kπ + θ)/n]

(c) r^1/n [cos(2kπ + θ)/n + i sin(2kπ + θ)/n]

(d) r^1/n [cos(2kπ + θ) – i sin(2kπ + θ)]

This question was posed to me in my homework.

Asked question is from Quadratic Equations topic in chapter Complex Numbers and Quadratic Equations of Mathematics – Class 11

Answer 1

Correct option is (c) r^1/n [cos(2kπ + θ)/n + i sin(2kπ + θ)/n]

The best I can explain: If n is any integer, then (cosθ + isinθ)^n = cos(nθ) + i sin(nθ).

Writing the binomial expansion of (cosθ + isinθ)^n and equating real parts of cos(nθ) and the imaginary part to sin(nθ), we get,

cos(nθ) = cos^nθ – ^nC2 cos^n-2θ sin^2θ + ^nC4 cos^n-4θ sin^4θ + ……….

sin(nθ) = ^nC1 cos^n-1θ sinθ – ^nC3 cos^n-3θ sin^3θ + ……….

If, n is a rational number, then one of the value of (cosθ + isinθ)^n = cos(nθ) + i sin(nθ).

If, n = p/q, where, p and q are integers (q>θ) and p, q have no common factor, then (cosθ + isinθ)^n has q distinct values one of which is cos(nθ) + i sin(nθ)

If, z^1/n = r^1/n [cos(2kπ + θ)/n + i sin(2kπ + θ)/n], where k = 0, 1, 2, ……….., n – 1.