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    The method I use here was heavily inspired (just like most of Beyoncé's songs) by the one outlined in Tristan Needham's Visual Complex Analysis, where he used a similar way to find the derivative of \tan{\theta}.

    So we start by constructing a right triangle of unit hypotenuse and angle \theta. Then the legs would be \sin{\theta} and \cos{\theta}. Then if we change the angle by \Delta\theta, keeping the hypotenuse constant of course, we get something similar to the attached diagram. \Delta\sin{\theta} and \Delta\cos{\theta} represent the change in \sin{\theta} and \cos{\theta} respectively, due to the change in the angle.
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    From the diagram, we can see that as \Delta\theta becomes infinitely small, the arc \Delta\theta approaches a straight line (dotted) and the radii (unit hypotenuses) become parallel. Not only that but also the arc/eventual straight line will become perpendicular to BOTH radii. So this small right triangle \bigtriangleup DCC' will eventually have hypotenuse \Delta\theta.

    Angles \theta and \angle ACD are congruent as they are alternate angles. And since \angle ACC' is right, m\angle CC'D = \theta. So we can easily deduce the following:
    \displaystyle \frac{\Delta\sin{\theta}}{\Delta \theta} \approx \cos{\theta} \implies \lim_{\Delta\theta\rightarrow 0} \frac{\Delta\sin{\theta}}{\Delta \theta} = \frac{d \sin{\theta}}{d \theta}=\cos{\theta} , and
    \displaystyle \frac{\Delta\cos{\theta}}{\Delta \theta} \approx -\sin{\theta} \implies \lim_{\Delta\theta\rightarrow 0} \frac{\Delta\cos{\theta}}{\Delta \theta} = \frac{d \cos{\theta}}{d \theta}= -\sin{\theta} .
    That negative sign comes from the fact that \cos{\theta} decreased here.
    Done!

    Hope you found it interesting!
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    (Original post by gagafacea1)
    The method I use here was heavily inspired (just like most of Beyoncé's songs) by the one outlined in Tristan Needham's Visual Complex Analysis,
    It's interesting to hear of a singer who draws on mathematics as the wellspring of her creativity. This is unusual and the only similar example with which I'm familiar was Elvis:

    https://groups.google.com/forum/#!se...4/Ufe9q2SXZ_wJ
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    (Original post by atsruser)
    It's interesting to hear of a singer who draws on mathematics as the wellspring of her creativity. This is unusual and the only similar example with which I'm familiar was Elvis:

    https://groups.google.com/forum/#!se...4/Ufe9q2SXZ_wJ
    lmaoooooo but actually meant that her songs are heavily inspired, not inspired by complex analysis. Though she does have "1+1" (song)!
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    (Original post by gagafacea1)
    Tristan Needham's Visual Complex Analysis
    This is an excellent book - it's my favourite on complex analysis.
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    (Original post by Brit_Miller)
    This is an excellent book - it's my favourite on complex analysis.
    FINALLY! :awesome: Somebody else who appreciates this book. I've always liked the idea of complex numbers, but this book just made me LOVE them. As a non-undergraduate, I found it really nice and easy to navigate without all the rigor that I don't have the time for.
 
 
 
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