De Moivre's Theorem Explained: Powering Complex Numbers Made Easy!

Want to learn how to raise complex numbers to a power easily? 🤔 De Moivre's Theorem is the key! In this video, I will break down this powerful theorem, explain why it works, and show you how to use it in trigonometry and complex number calculations.
📌 What You'll Learn:
✔️ What is De Moivre's Theorem?
✔️ How to apply it to complex numbers in polar form
✔️ Solving powers and roots of complex numbers step-by-step
✔️ Example problems to reinforce your understanding
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[@aryusure1943]

Well, that's where I gave up after high school and chose humanities without math at the CEGEP (2 years of studies in preparation for University) level (province of Québec, Canada). I quit when I was ahead. ;) So, this stuff is not something I learned to begin with. I remember vaguely trigonometry but I need to refresh my notions a lot to get to this level of math.

[@terry_willis]

I just read the wiki bio of de Moivre. This guy was a freakin' genius. 😵‍💫

[@BrenoStrato]

For those who wish to remember: r² = a² + b². θ = arcsin (b/r) and arccos (a/r) .

[@warblerab2955]

unfortunately, I didn't get this far in math. I have no idea how to take complex number and put it in polar form. :(

[@Teronix100]

*T = Theta* It's triangles and circles. Think of a circle on an x,y plane. It has radius 1. The x axis ~ cos(T); the y axis ~ sin(T). If you make a point on that circle you can make a triangle starting at 0,0. The hypothenuse is 1, the sides are sin(T) and cos(T). If you want to make any number, you could describe it with a circle from the origin and the angle where to look. OR You can describe it with x,y coordinates, but those are equivalent to triangle lengths. If you want to go from x,y to polar. First we need to do the radius It is important to note you need the lenght of those, not the actual value. i has a lenght of 1 in the unit circle. So instead of i multiply with 1. Negative numbers are just a direction so neg. can be skipped, but it doesn't matter. They are important to determine the angle, not the radius Since the radius is the hypotenuse you can use phytagoras. In that case knowing i=1 it is r² = (√3)² + (1) ² = 3 +1 = 4; So r =2 Now you can use your favourite arcsin or arccos to get Theta; It doesn't matter. In that either 180 - arcsin(1/2) or arccos(-√3/2). Both will give you the angle for both sin(T) and cos(T). This will yield you either 150°. The 180 is there because the angle starts at x,y = 1,0 => Theta = 0 against the clock. arcsin(1/2) = 30°. So you have to keep in mind, where you are in the 4 Quadrants of the x,y plane and if the number you get is sensible. sin(150°) = sin(30°) = 1/2

[@panlomito]

Stepping from high school to MIT...

[@JohnDoe-ok5wo]

I don't get how theta became 150 degrees.

[@BrenoStrato]

r² = a² + b². θ = arcsin (b/r) and arccos (a/r) . If you want to better understand how we arrived at these formulas, sketch the complex plan. The horizontal axis corresponds to the real part and the vertical axis corresponds to the imaginary part. Then use trigonometry to deduce.