Question

Use Demoivres Theorem to find (1 + i) 20.
a. 1024i
b. -1024
C.-1024
D.1024

Answer 1

C on e2020

i did it *dab*

Answer 2

-1024

Explanation:

Moivre's theorem allows to easily obtain trigonometric formulas that express the sine and cosine of a multiple angle as a function of the sine and cosine of a simple angle.

De Moivre's theorem can be applied to any complex number
`z`

Where:

`z\in Z`

Let:

`(1+i)=z\\n=20`

According to Demoivres Theorem, If:

`z=||z||(cos(\theta)+isin(\theta))`

Then:

`z^n=||z||^n(cos(n\theta)+isin(n\theta))`

For a complex number
`z`:

`z=a+bi`

Its magnitude and angle are given by:

`||z||=√(a^2+b^2) \\\\\theta=arctan((b)/(a) )`

So:

`||z||=√(1^2+1^2) =√(2)`

`\theta=arctan((1)/(1) )=45^(\circ)`

Therefore, using De Moivre's theorem:

`z^n=(√(2) )^(20)(cos(20*45)+isin(20*45))\\\\z^n=(√(2) )^(20)(cos(900)+isin(900))\\\\z^n=1024(-1+i(0))\\\\z^n=1024(-1)\\\\z^n=(1+i)^(20)=-1024`