- [Voiceover] So in this video we're goint to talk about
Euler's formula.
And one of the things I want to start out with is
why do we want to talk about this
rather odd looking formula?
What's the big deal about this?
And there is a big deal.
And the big deal is e.
We love e.
And I'll underline that twice.
Now the reason is, because
when we take a derivative of e
D DT
of e to the x
equals e to the x.
And D DT
of e to the ax
so where a is anything
equals ae to the ax.
And so the property is that when you take a derivative
of the function, the same function comes out.
Or, if you take a derivative of the function
a scaled version of the same function comes out.
And we love this, because why?
Because when we do differential equations
e to the x
is the solution.
Almost every time.
Whenever we did a circuit
e to the x was the answer.
If you recall from when we were solving
circuit simple circuits with differential equations
that we always said something like
well we're gonna guess that V of T
is some constant times e to the
st.
That was a proposed solution.
This turned out to work every time.
So there's something else we love, too.
And that is
sinusoids.
Or, sines and cosines.
Okay.
We love these.
And that gets two lines.
Now why do we love these?
Is because they happen in nature.
If you whistle
the air pressure looks like a sine wave.
If you ring a bell
the bell moves in a sine wave.
In any kind of music
if you look at the notes in music
the sound they make
the pressure waves look like
sine waves.
And circuits make sine waves, remember?
We analyzed this circuit in great detail
it was the
LC circuit.
We looked at the natural response of this
and that was a sine wave.
Okay.
So, electric circuits make sine waves.
All these things make sine waves.
They occur in nature.
And we want to be able to analyze
things that happen when sine waves are present.
So, we have two things we love and we want
to relate these two things.
And these are going to be related
through that.
Euler's Formula.
That's how we connect these two separate ideas.
Let me, let's go do that.
So Euler's Formula says that
e to the jx
equals
cosine
X
plus
j
times sine x.
Sal has a really nice video where he actually
proves that this is true.
And he does it by taking the MacLaurin
series expansions of e,
and cosine, and sine
and showing that this expression is true
by comparing those series expansions.
And I'm not going to repeat that here
we're just going to state that as fact.
And now we're going to look at this equation a little bit more.
So, this is the expression that relates exponentials
that we love, to sines and cosines that we love.
And part of the price of doing that is we introduce
complex numbers into our world.
Here's two complex numbers.
Okay.
This is where complex numbers come into
electrical engineering.
So we have to mention the other form of
this formula which is
e to the, I put a minus sign in here,
e to the minus jx.
And that equals
cosine X
minus J
sine X.
So these two expressions together
are Euler's Formula, or Euler's Formulas.
And we're gonna exploit this by taking
we'll be able to take the cosines and sines
that we find in nature
we're going to be able to fashion them into exponentials.
And these exponentials then go into our
differential equations and give us solutions.
And we're going to come back and pull out the cosines
and sines.
That's the rhythm of how we're going to use this equation
to help us solve circuits.
One small point I want to share,
notice that in both these equations the sine comes first?
And the sine is over here
on the imaginary side.
So the cosine is the real side.
This is the reason that we have a preference in the future
we're going to have a preference
if we're talking about our real world signals
in terms of the cosine function.
It's because in this Euler's Formula
the cosine comes first, in both cases.
So what I want to do now is
take a second and I want to see
if we had our signal expressed in these exponentials
how do we recover the cosine and the sine term?
How do we flip these equations around
so we can solve for the cosine and the sine?
It's a simple bit of algebra here.
It's good to see.
All right, so if I want to isolate the cosine term.
If I want to isolate the cosine term
let me get rid of these guys here.
So now to isolate the cosine term
what I'm going to do is add these two equations together
and that plus and minus are going to cancel out
this second term here.
That's what I'm going for.
So, if I add
I'll get e to the jx
plus
e to the minus jx
equals, cosine doubles,
two cosine X.
And the two sine terms cancel out.
All right?
So then I could write
I'll write it over here.
I can write, cosine of X
equals
e to the plus jx
plus e to the minus jx
over two.
All right?
So that's the expression
that's the expression for cosine in terms of
complex exponentials.
Okay, let's go back and see if we can get sine.
So what I'm gonna do with sine
is I'm going subtract.
And that gives me
what that'll do is that'll get the cosine terms
to fall away.
And then I get
e to the jx
subtracting, right?
Minus e to the minus jx
equals
cosine terms, they subtract out.
And I get two times
j
times sine X.
All right, and that means I can write
sine X
equals
e to the plus jx
minus e to the minus jx
over two j.
And it's really easy
that j term is down there.
That's easy to forget sometimes.
So I put a square around these guys
'cause that's important.
So there's the two expressions.
That's the two expressions for
if you have complex exponentials and you want
to extract the cosine,
this is how you do it.
And if you want to extract the sine
this is how you do it.
Okay?
And you can either
you can put these in your head or
probably easier for me,
if you just remember
Euler's Formula, this is a pretty straightforward
quick derivation.
So the other thing we put a square around
is this guy here.
So we have Euler's Formula.
And basically the cosine and sine extracted
from Euler's Formula.