Simplifying Complex Numbers Using Euler's Formula

[GreenPrint]

Homework Statement

Even if you have no idea what on Earth any of this stuff is but have a vast knowledge in complex number system you might be able to help me as this is what I believe is preventing me from making sense of this stuff...

Ok I'm trying to understand how you would know that sin(pi/4) is equal to sqrt(2)/2 if you were never told that it was so... not doing any gemoetric proofs... yes I know there are some for those angles... instead I would like to turn to works of euler and try to understand that way...

According to Euler... yes this question has an answer as the formula has been around for a couple hundred of years and somebody should be help me understand...

sin(x) = ( e^(ix) - e^(-ix) )/(2i) or solemly noted ever as ( cis(x) - cis(-x) )/(2i) which one you prefer
I was woundering if somebody as well could tell me what cis stands for
like you know how sin(x) is spoken sine of x what is cis
I thought it might be like coimaginarysine or something
I have always been told that cis(x)= e^(ix) = meh don't feel like fliping thorugh my notebook but you you should know what that is also equal to and was never told how to pronounce cis

fine
e^(ix) = cos(x) = cosx + i sin(x)
e^(-ix) = cos(x) - i sin(x) = 1/e^(ix)

anways inspection formula yield me to come to the conclusions that sense in the formlua
( e^(ix) - e^(-ix) )/(2i)
were dividing by 2 here the numerator should yield the length of the unit circle that the sine of the angle refers to times 2 times i

sure enough pluging into your calculator the numerator the formula with x being pi/4 you get some number and putting in isqrt(2) you get exactly the same just as I had sepeculated, twice the length on the unit circle coresponding to that angle times the imaginary number because sin is a reference to the second dimension, the set of complex numbers... hence dividng that number by 2i you get none other then sqrt(2)/2 which makes perfect sense...

now my question is how would I know what the numerator equal isqrt(2) if I didn't have my calculator how do I simplify the numerator pluging in pi/4 for x and getting isqrt(2) I was woundering if you could show me how to do this...

this may help refresh your memory on some things

e^(ipi) = -1
ipi = ln(-1)
pi = ln(-1)/i
i = ln(-1)/pi = sqrt(-1)

as I stated early this stuff has been around for several centuries especially the complex number system which my lack of knowledge is hindering my ability to simplify the numerator which I am hopeing somebody on here could help me with

Also one other thing that spooked me was if pluging pi/4 into this equation

e^(ix) - e^(-ix) = isqrt(2)
and I divide through by i I have the exact value of sqrt(2)... why? Can someone please enlighten me on this as well... what is this deffintion of square rooting a number never seen it ever... sure rasing a numer to some other number and rasing that number to -1, i.e. 2^(2^-1) = 2^(1/2) = sqrt(2) or simply what times waht equals two as taught in like six grade put what on Earth is this deffintion?

Well hopefully someone will be able to help me as this stuff has been around for a good of chunck of time and someone has to have poundered this stuff before me especially euler when he derrived the stuff... so although there might not be many people that can hopefully someone on here will be able to help...

THANKS MILLIONS!

Homework Equations

The Attempt at a Solution

 

[Raskolnikov]

I was woundering if somebody as well could tell me what cis stands for
like you know how sin(x) is spoken sine of x what is cis

cis(x) is just another name for the complex exponential function e^(ix). To the best of my knowledge, it's pronounced as "sis x."

sure enough pluging into your calculator the numerator the formula with x being pi/4 you get some number and putting in isqrt(2) you get exactly the same just as I had sepeculated, twice the length on the unit circle coresponding to that angle times the imaginary number because sin is a reference to the second dimension, the set of complex numbers... hence dividng that number by 2i you get none other then sqrt(2)/2 which makes perfect sense...

now my question is how would I know what the numerator equal isqrt(2) if I didn't have my calculator how do I simplify the numerator pluging in pi/4 for x and getting isqrt(2) I was woundering if you could show me how to do this...

I think you're making things way too complicated.

$$\frac{e^{ix} - e^{-ix} }{2i}$$

$$= \frac{(cos(x) + i*sin(x)) \ - \ (cos(-x) + i*sin(-x))}{2i}$$

$$= \frac{(cos(x) + i*sin(x)) \ - \ (cos(x) - i*sin(x))}{2i}$$

$$= \frac{2i*sin(x)}{2i}$$

$$= sin(x).$$

This is the proof for the general case. It doesn't matter what x you choose.

 

[GreenPrint]

I know the proof lol but what about the numerator stuff that I mentioned above... kind of confused me

 

[Raskolnikov]

What about the numerator?

Also one other thing that spooked me was if pluging pi/4 into this equation

e^(ix) - e^(-ix) = isqrt(2)
and I divide through by i I have the exact value of sqrt(2)... why? Can someone please enlighten me on this as well... what is this deffintion of square rooting a number never seen it ever... sure rasing a numer to some other number and rasing that number to -1, i.e. 2^(2^-1) = 2^(1/2) = sqrt(2) or simply what times waht equals two as taught in like six grade put what on Earth is this deffintion?

I think you're not realizing that all this is connected. The reason you were never taught that "definition" is because we build up on what we know as we learn. It'd be silly to learn this before learning Euler's Formula...before learning complex numbers...before learning trig...before learning exponents...etc.

 

[Raskolnikov]

There are a lot of cool results you run into just by manipulating different equations. For example:

$$i^i = [cos(\pi/2) + i*sin(\pi/2)]^i = (e^{\frac{i\pi}{2}})^i = e^{\frac{-\pi}{2}}.$$

This means an imaginary number raised to an imaginary power is actually a real number! That's pretty neat and unexpected when you go through it that backwards way of "result-first, derivation-later." But if we were to take the forward approach starting with the general Euler's Formula, it'd make more sense.