The Millionth Roots of i - de Moivre's theorem

[TiPster]

Find the millionth roots of i and plot your answer.

I'm finishing that last two chapters of trig on my own over the summer and this one has stumped me for the past few weeks.

I know that the resulting graph will be a circle (or technically, a million-sided shape, but close enough). I know I'm supposed to change the algebraic form (i) to the polar form (this is what I'm having trouble with).

I'm supposed to make a graph with a triangle, but the end result I keep getting is 0cos(180)+0isin(180). Obviously, this results in 0.
This is the graph form I'm supposed to be using:

Then I write it in polar form: [FONT=MathJax_Math]x[FONT=MathJax_Main]+[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Math]j[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]r[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]cos[/FONT][FONT=MathJax_Math]θ[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Math]j[/FONT][FONT=MathJax_Main]sin[/FONT][FONT=MathJax_Math]θ[/FONT][FONT=MathJax_Main])[/FONT][/FONT]

Then, if I get the polar form, I get to move on to de Moivre's theorem and I can't even understand how that factors in.

Thank you so much if you can help me. I hope I've explained what I have so far well.

 

[Deleted member 4993]

I'm finishing that last two chapters of trig on my own over the summer and this one has stumped me for the past few weeks.

I know that the resulting graph will be a circle (or technically, a million-sided shape, but close enough). I know I'm supposed to change the algebraic form (i) to the polar form (this is what I'm having trouble with).

I'm supposed to make a graph with a triangle, but the end result I keep getting is 0cos(180)+0isin(180). Obviously, this results in 0.
This is the graph form I'm supposed to be using:
View attachment 3028
Then I write it in polar form: [FONT=MathJax_Math]x[FONT=MathJax_Main]+[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Math]j[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Math]r[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]cos[/FONT][FONT=MathJax_Math]θ[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Math]j[/FONT][FONT=MathJax_Main]sin[/FONT][FONT=MathJax_Math]θ[/FONT][FONT=MathJax_Main])[/FONT][/FONT]

Then, if I get the polar form, I get to move on to de Moivre's theorem and I can't even understand how that factors in.

Thank you so much if you can help me. I hope I've explained what I have so far well.

You need to make:

\(\displaystyle r \ = \ \sqrt{x^2 + y^2}\)...................from the triangle you drew and using Pythagorean theorem

\(\displaystyle \displaystyle \theta \ = \ \tan^{-1}\left (\frac{y}{x}\right )\)

for example you want to convert 1 + i to polar form. Then we have x = 1 and y = 1 and

r = √(1² + 1²) = √2

and

Θ = tan^-1(1/1) = tan^-1(1) = π/4

Then

1 + i = √2 * [cos(π/4) + j sin(π/4)]

That's it....

 

[Deleted member 4993]

Finiding millionth root of i is different problem

There you need use:

x + i y = r * e^iΘ

You have:

i = e^i*π/2

(i)^1/1000000 = e^i*π/2000000 = cos(π/2000000) + i * sin(π/2000000) = 0.999999999998766 + i * 1.5701*10^-6

 

[TiPster]

Wow, thank you! That made a lot more sense.