Let z = −2√3 + 6ia) Express z|z| in polar form.b) Determine z48 in polar form, in terms of its principal argument.c) Determine all distinct values of z1/4 in polar form, in terms of their principal argument.
Question
Let z = −2√3 + 6ia) Express z|z| in polar form.b) Determine � z48 in polar form, in terms of its principal argument.c) Determine all distinct values of z1/4 in polar form, in terms of their principal argument.
Solution
a) To express z in polar form, we first need to find the modulus and argument of z.
The modulus of z, |z|, is given by √((Re(z))^2 + (Im(z))^2) = √((-2√3)^2 + (6)^2) = √(12 + 36) = √48 = 4√3.
The argument of z, arg(z), is given by arctan(Im(z)/Re(z)) = arctan(-6/(-2√3)) = π/6.
So, z = |z|(cos(arg(z)) + isin(arg(z))) = 4√3(cos(π/6) + isin(π/6)).
b) To find z^4/8 in polar form, we use De Moivre's theorem, which states that (r(cos θ + i sin θ))^n = r^n (cos(nθ) + i sin(nθ)).
So, z^4/8 = (4√3)^4/8 * (cos(4π/6) + isin(4π/6)) = 324(cos(2π/3) + isin(2π/3)).
c) To find the fourth roots of z, we use the formula z^(1/n) = r^(1/n) (cos((θ+2kπ)/n) + i sin((θ+2kπ)/n)), where k = 0, 1, ..., n-1.
So, the fourth roots of z are:
z^(1/4) = (4√3)^(1/4) * (cos((π/6 + 2kπ)/4) + i sin((π/6 + 2kπ)/4)), for k = 0, 1, 2, 3.
Calculating these gives the four roots:
- √2(cos(π/12) + isin(π/12))
- √2(cos(5π/12) + isin(5π/12))
- √2(cos(9π/12) + isin(9π/12))
- √2(cos(13π/12) + isin(13π/12))
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