Question

Find the complex fourth roots of 81(cos(3π/8)+isin(3π/8)). a) Find the fourth root of 81. b) Divide the angle in the problem by 4 to find the first argument. c)Use the fact that adding 2π to the angle 3π/8 produces the same effective angle to generate the other three possible angle for the fourth roots. d) Find all four of the fourth roots of 81(cos(3π/8)+isin(3π/8)). express your answer in polar form.

Answer 1

The answer is below

Explanation:

Let a complex z = r(cos θ + isinθ), the nth root of the complex number is given as:

`z_1=r^{(1)/(n) }(cos((\theta +2k\pi)/(n) )+isin((\theta +2k\pi)/(n) )),\\k=0,1,2,.\ .\ .,n-1`

Given the complex number z = 81(cos(3π/8)+isin(3π/8)), the fourth root (i.e n = 4) is given as follows:

`z_(k=0)=81^{(1)/(4) }(cos(((3\pi)/(8) +2(0)\pi)/(4) )+isin(((3\pi)/(8) +2(0)\pi)/(4) ))=3[cos((3\pi)/(32) )+isin((3\pi)/(32))] \\z_(k=0)=3[cos((3\pi)/(32) )+isin((3\pi)/(32))]\\\\z_(k=1)=81^{(1)/(4) }(cos(((3\pi)/(8) +2(1)\pi)/(4) )+isin(((3\pi)/(8) +2(1)\pi)/(4) ))=3[cos((19\pi)/(32) )+isin((19\pi)/(32))] \\z_(k=1)=3[cos((19\pi)/(32) )+isin((19\pi)/(32))]\\\\`

`z_(k=2)=81^{(1)/(4) }(cos(((3\pi)/(8) +2(2)\pi)/(4) )+isin(((3\pi)/(8) +2(2)\pi)/(4) ))=3[cos((35\pi)/(32) )+isin((35\pi)/(32))] \\z_(k=2)=3[cos((35\pi)/(32) )+isin((35\pi)/(32))]\\\\z_(k=3)=81^{(1)/(4) }(cos(((3\pi)/(8) +2(3)\pi)/(4) )+isin(((3\pi)/(8) +2(3)\pi)/(4) ))=3[cos((51\pi)/(32) )+isin((51\pi)/(32))] \\z_(k=3)=3[cos((51\pi)/(32) )+isin((51\pi)/(32))]`