Nate's blog » sin http://www.natenewz.com My various projects Tue, 29 Jun 2010 18:00:42 +0000 http://wordpress.org/?v=2.9 en hourly 1 Euler’s Formula and Trig Identities http://www.natenewz.com/2009/12/16/eulers-identity-and-trig-identities/ http://www.natenewz.com/2009/12/16/eulers-identity-and-trig-identities/#comments Wed, 16 Dec 2009 06:12:32 +0000 nate http://www.natenewz.com/?p=250 Electrical Engineers tend to use imaginary numbers quite often for various reasons. The term ‘imaginary’ can be misleading. In algebra II, when I heard that term, I was skeptical about the usefulness of an imaginary quantity. But if you can get over the name, they can be used as a powerful tool in many math, and physics problems. I took a whole class on complex variable analysis in the spring of 09′ after I realized how important they were to electrical engineering problems. Anyways, to illustrate one simple use of a complex number, I thought I would derive a trig formula. This technique comes in handy if you ever forget a trig formula, and you need to figure it out quick.
sin(x)^{3}=?
sin(x)=\frac{e^{ix}-e^{-ix}}{2i}– Euler’s Identity
y=e^{ix}– a Variable substitution to simplify things
sin(x)^{3}=(\frac{y-1/y}{2i})^{3}
=\frac{({y-1/y})^{3}}{(2i)^3}=\frac{y^3-3y+\frac{3}{y}-\frac{1}{y^3}}{(2i)^3}
=\frac{y^3-1/y^3}{(2i)^3}-3\frac{y-1/y}{(2i)^3}
i*i=-1
(2i)^3=-4*2i
=-\frac{1}{4}\frac{y^3-1/y^3}{2i}+\frac{3}{4}\frac{y-1/y}{2i}
=-\frac{1}{4}\frac{e^{i3x}-e^{-i3x}}{2i}+\frac{3}{4}\frac{e^{ix}-e^{-ix}}{2i}
sin(x)^3=-\frac{1}{4}sin(3x))+\frac{3}{4}sin(x)
two functions graphed on top of each other

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