Find all the complex roots. Leave your answers in polar form with the argument in degrees. The c

Find all the complex roots. Leave your answers in polar form...

Key Concepts

Polar Representation of Complex Numbers

Polar representation expresses a complex number in terms of its magnitude (or modulus) and angle (or argument), typically written as r(cos(?) + i sin(?)) or in exponential form as re^(i?). It makes operations such as multiplication, division, and root extraction more straightforward, particularly when dealing with rotations and scalings in the complex plane.

Euler's Formula

Euler's formula, e^(i?) = cos(?) + i sin(?), creates a powerful connection between trigonometric functions and exponential functions. This identity is fundamental in converting between polar and exponential forms, which simplifies many operations on complex numbers, including finding roots.

De Moivre's Theorem

De Moivre's theorem states that for any complex number in polar form, [r(cos(?) + i sin(?))]^n = r^n [cos(n?) + i sin(n?)] for any integer n. This theorem is essential for raising complex numbers to powers and extracting n-th roots, by allowing the exponentiation of the angle as well as the magnitude.

n-th Roots of Complex Numbers

The process for finding the n-th roots of a complex number involves determining all n distinct solutions that satisfy the equation z^n = w, where w is given in polar form. This is done by taking the n-th root of the magnitude and dividing the argument by n, while accounting for the periodicity of angles by adding multiples of 360°/n (or 2?/n in radians) to generate all distinct roots.

Transcript

00:01 So i'm looking for the five roots or the fifth complex roots for negative i.

00:07 So we're looking for negative i to the one -fifth power.

00:12 Now the easiest way to work with this is to put this into polar form.

00:16 So first let's sketch a little graph.

00:18 So negative i is right here with a modulus of one because one unit away from the origin and it has a rotation of 270 degrees.

00:27 So 1, cis 270 degrees to the 1 fifth power would be the equivalent of negative 1 to the 1 5th power.

00:37 Now, for those that don't know, cis 270 is a shortcut or abbreviation for the cosine of 270 degrees plus i sign of 270 degrees.

00:48 So i'm just using this to shortcut so we don't have to write as much.

00:52 Now, in order to work this problem, the first thing that i need to do is i need to break up 360, which is one rotation into five pieces.

01:00 This will tell us how much we need to rotate between answers to get all five answers within the one cycle.

01:07 Now, this won't tell us where to start, but it will tell us how far in between each rotation will be.

01:13 So 360 divided by 5 is 72 degrees.

01:17 Okay, so we're going to put a pin in that for a quick second.

01:20 Now over here, i'm going to use demov's theorem.

01:23 One to the one -fifth power will give us the modulus of all of our answers.

01:27 So i'm going to write this in formal notation.

01:30 So one, two, three, four, five answers, all with a modulus of one.

01:38 And i won't use the shorthand for the final answer.

01:43 Cosine plus i sine, cosine plus i sign, cosine plus i sign, cosine plus i sign, cosine plus i sign, cosine plus i sign, cosine plus i sign...

Please give Ace some feedback