Question

I don't understand DeMoivre's theorem too well, nor do I feel it will be incredibly important, in PreCal or even Calculus. I have a specific question therefore that I could really use some help with:
Find (1 + i)^8 using DeMoivre's Theorem.
I guess it wants me to write it in standard form. This thing does a god awful job of explaining the instructions clearly.
Also, if you've taken PreCal or Calculus, can you please let me know how important this theorem is? Thanks!!

Answer 1

ok here is what we need to do, because demoivre is easy to implement
first we have to write \(1+i\) in trigonometric form
do you know how to do that?

Answer 2

guess not
oh well...

Answer 3

the idea is to write \(i+i\) in trig form , which is straightforward enough. you need to write it as
\[1+i=r(\cos(\theta)+i\sin(\theta))\] then raising it to the power of 8 will be a snap
take \(r^8\) and multiply \(\theta\) by 8
that is why it is so easy, just multiply the angle

Answer 4

in order to turn \(1+i\) in to \(r(\cos(\theta)+i\sin(\theta))\) you need to numbers, \(r\) and \(\theta)\)
\(r=\sqrt{a^2+b^2}\) in your case \(r=\sqrt{1^2+1^2}=\sqrt{2}\)
\(\theta \) is the angle where \(\tan(\theta)=\frac{b}{a}\) so in your case \(\tan(\theta)=1\) and so \(\theta=\frac{\pi}{4}\)

Answer 5

*two numbers

Answer 6

therefore \(1+i=\sqrt{2}(\cos(\frac{\pi}{4})+i\sin(\frac{\pi}{4}))\)
this in an equality that is easy to verify

Answer 7

now raising it to the power of 8 is easy
\(\sqrt{2}^8(\cos(\frac{\pi}{4}\times 8)+i\sin(\frac{\pi}{4}\times8))\)

Answer 8

last job is to evaluate this number, and i will leave that up to you

Answer 9

Thanks !! :D

Answer 10

yw

Answer 11

you will use this theorem A LOT in calculus. To be honest I didn't even know it was a theorom, but pretty much every time you use eulers formula you will also use de Moivre's.

Answer 12

well I guess not pretty much everytime, but a bit still

Answer 13

I forget my trig functions a lot, and Calc uses a lot of trig functions. I use it to derive the different identities/formulas from Euler's formula with de Moivre's

Answer 14

can pretty much derive almost all of your trig identities by writing \(\cos(\theta)+i\sin(\theta)=e^{i\theta}\) , using the regular old laws of exponents, and then equating real and imaginary parts
for example you can prove the "addition angle" formula for sine and cosine using this