# Exponential and trigonometric relation

#### Mathslover123

##### New member
Is this relation correct ?
Actually this is the problem in physics but its related to maths more so...
Is e^-ix = sinx ?
I found this in several derivations.
If i understand this , its gonna be a lot easier to begin with a lot of derivation of simple harmonic motion ...

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#### MarkFL

##### Super Moderator
Staff member
Euler tells us:

$$\displaystyle e^{i\theta}=\cos(\theta)+i\sin(\theta)=\cis(\theta)$$

Yes sir..

#### pka

##### Elite Member
Euler tells us: $$\displaystyle e^{i\theta}=\cos(\theta)+i\sin(\theta)=\cis(\theta)$$
Yes sir..
What kind of answer is that?
Mathslover123, are you trolling us on this site?
Lets hope not. Thus $$\displaystyle e^{-ix}=\cos(-x)+i\sin(-x)=\cos(x)-i\sin(x)$$
Now what is your question or point?

#### Mathslover123

##### New member
What kind of answer is that?
Mathslover123, are you trolling us on this site?
Lets hope not. Thus $$\displaystyle e^{-ix}=\cos(-x)+i\sin(-x)=\cos(x)-i\sin(x)$$
Now what is your question or point?
No no no... I aint trolling. I am expecting an answer to my question.I have a problem in finding general solutions to differential equations of simple harmonic motion ( mainly damped and forced ) .

#### Mathslover123

##### New member
Does my replies sound rude or disrespectful ? You always ask me about what kind of answer was that ? Sir,i think i am here to find an answer to my questions...because my teachers at college really bad in doing it...otherwise why would i be here instead of asking to whom i have paid !

#### Dr.Peterson

##### Elite Member
Yes, some of us are being a little rude, but I can see why.

I think the problem is that your question doesn't make sense to us, and your answer ("yes sir", with no further comment or question) suggests you aren't thinking. So we don't know what to say next.

You asked "Is e^-ix = sinx ?"

You were told, "$$\displaystyle e^{-ix}=\cos(-x)+i\sin(-x)=\cos(x)-i\sin(x)$$". That is, no, it is not even a real number!

Now you need to tell us more, and/or ask us more.

The material you are using appears to be saying nonsense, unless we are missing something important! The fact is that they can't replace sin(u) with e^-iu, unless perhaps there is something hidden (you showed only small part of a page) that would show what they are really doing. We need to know that context. If your question arose because you are confused, then so are we ... (at least I am).

Can you at least show us more of what is being said there, and tell us about the overall context? My guess is that something is being done such that only the imaginary part matters (I would have expected it to be the real part), and what you showed doesn't mention that.

#### Mathslover123

##### New member
I thought the same. It must have takem real or imaginary part.But what suggests us to do so.
Is it because the LHS of eq 17.20 represents real quantity , due to which the imaginary part of RHS becomes zero ? So that we will be finally left with equation having sine.

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#### Dr.Peterson

##### Elite Member
It would have been a good idea to show us the original to start with!

He says, "In exponential form this equation can be expressed as ...", in explaining the step you asked about. I suspect that something has previously been said about what "exponential form" means in this context, and that it involves creating a new equation whose real part (or imaginary part?) agrees with the given equation. Look for that.

The context agrees with what I was expecting, namely quantum mechanics. (Context is always important in asking a question.) Complex functions whose real part represents observable reality are typical there. (That's about all I know about quantum mechanics ...) So I expect you to have been told about that previously.

And it looks like the incomplete sentence at the bottom of what you showed us may be about to say something about the significance of real and imaginary parts!

(I have to say that there is enough odd grammar or word choice there - "To peruse"? - so that I find it hard to follow; but maybe more context would help.)

#### Mathslover123

##### New member
Sir is this logical ?? Please check this..

#### Dr.Peterson

##### Elite Member
Surely you don't believe that all solutions of a differential equation are equal! These are just two different members of the family of solutions.

The point I tried to make last time was that the author can't be claiming that the two are equal (though that is what he seems to be saying). When he says "exponential form" he must mean something else that he has previously discussed. If I were working with you face to face, I would have long ago taken the book from you and started looking through it for mentions of this "exponential form" and of the significance of the real or imaginary parts of a wave function. Is there an index entry for "exponential form"?

I am very much hoping a physicist will jump in here; but it's quite likely that the presentation you have been given is not standard.

#### Mathslover123

##### New member
No there is no any mentioning about exponential form. The author has directly used that thing. May be i should just remember that in my mind rather than searching for its logic.