Question

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

Answer 1

Answer:

-1024

Step-by-step 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:

$||z||=\sqrt{a^2+b^2} \\\\\theta=arctan(\frac{b}{a} )$

So:

$||z||=\sqrt{1^2+1^2} =\sqrt{2}$

$\theta=arctan(\frac{1}{1} )=45^{\circ}$

Therefore, using De Moivre's theorem:

$z^n=(\sqrt{2} )^{20}(cos(20*45)+isin(20*45))\\\\z^n=(\sqrt{2} )^{20}(cos(900)+isin(900))\\\\z^n=1024(-1+i(0))\\\\z^n=1024(-1)\\\\z^n=(1+i)^{20}=-1024$

Answer 2

C on e2020

i did it *dab*