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B What's wrong with this proof of sin(i)=0?

  1. Nov 1, 2016 #1
    We have,
    e^(ix)=cosx+isinx
    So, e^(i*i)=cosi+isini
    Or e^-1=cosi+isini
    Or 1/e + 0*i= cosi+isini
    So, cosi=1/e and sini=0
    But that's not the value of sin(i) that I found on the internet. These values are not even satisfying cos^2(x)+sin^2(x)=1.
    What did I miss?
     
  2. jcsd
  3. Nov 1, 2016 #2

    fresh_42

    Staff: Mentor

  4. Nov 1, 2016 #3

    robphy

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    Gold Member

    e^-1=cosi+isini
    1/e + 0*i= cosi+isini

    On the left, you have the real and imaginary parts of ##e^{-1}##.
    So, you should find the real and imaginary parts on the right-hand side.
    Since ##\cos(i)=\cosh 1## and ##\sin(i)=i\sinh(1)##
    http://www.wolframalpha.com/input/?i=cos(i)
    http://www.wolframalpha.com/input/?i=sin(i)
    we have:
    ##\begin{align*}
    1/e + 0i
    &= \cosh 1+i (i\sinh 1)\\
    &= \cosh 1-\sinh 1\\
    &= \frac{e^1+e^{-1}}{2}-\frac{e^1-e^{-1}}{2}\\
    &= e^{-1}
    \end{align*}##
     
  5. Nov 1, 2016 #4

    lurflurf

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    This is ok

    We have,
    e^(ix)=cosx+isinx
    So, e^(i*i)=cosi+isini
    Or e^-1=cosi+isini
    Or 1/e + 0*i= cosi+isini

    This is false
    So, cosi=1/e and sini=0

    in fact
    cosi=cosh1
    isini=sinh1
    which are both real

    But that's not the value of sin(i) that I found on the internet. These values are not even satisfying cos^2(x)+sin^2(x)=1.
    What did I miss?

    what value did you find? The correct values I gave above satisfy cos^2(x)+sin^2(x)=1.
     
  6. Nov 1, 2016 #5
    So, the problem was when I compared both sides considering cosi and sini to be real, right?
    One more thing, does the sine of complex numbers have any physical meaning? Just like how the sine of real numbers represents the y-co-ordinate of the point that we reach on a unit circle by rotating throught that angle. Do they mean anything or are they just useless numbers obtained by plugging in complex numbers in the tailer series?
     
  7. Nov 1, 2016 #6

    mfb

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    2016 Award

    Staff: Mentor

    Right. Cosine and sine are complex in general - only for real arguments they are real.

    I'm not aware of physical applications of complex arguments for the sine. Damped oscillations can be described with a complex period in the exponential, but converting that to a sine and cosine does not really help.
     
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