For those who don’t know, and correct me if I’m wrong, Euler’s Formula is e^(xi) = cos(x) + isin(x) and Euler’s Identity is where “x” is equal to pi, therefore e^(πi) = -1. It’s fascinating to me how something so complex can be equal to something so simple (talking about Euler’s Identity). But what I don’t get is how they’re practical. What are they used for, and why are they important?
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Let me copy a comment I mad recently in another thread here:
Already the formula
e^(ix) = cos(x) + i·sin(x)
can make your life much easier. For example,
e^(ix) · e^(iy) = e^(i·(x+y))
by a simple power law. But expand that using the formula you get
(cos(x) + i·sin(x)) · (cos(y) + i·sin(y)) = cos(x+y) + i·sin(x+y).
Now expand the left hand side and compare real/imaginary parts and you get
cos(x+y) = cos(x)·cos(y) – sin(x)·sin(y),
sin (x+y) = cos(x)·sin(y) + sin(x)·cos(y).
Hence the rather complicated two(!) laws of addition for (co)sine are the same as a simple power law when described via complex numbers.
Another consequence of the formula is
cos(x) = (e^(ix) + e^(-ix)) / 2,
sin(x) = (e^(ix) – e^(-ix)) / 2i.
This not only tells you how to define (co)sines of absolutely any complex number x, but also give you a way to calculate arcsin and arccos! For example, setting y = e^(ix) and thus x = log(y)/i, lets calculate an x such that cos(x)=a:
By the formula, we want 2a = e^(ix) + e^(-ix) = y + 1/y, i.e. after multiplying by y and getting everything to one side y² – 2a·y +1 = 0. The formula for quadratic equations now tells you
y = a ± sqrt(a²-1)
and hence by x = log(y)/i we arrive at
arccos(a) = x = log(a ± sqrt(a²-1)) / i.
Lastly, you can combine the formula(s) with the one for geometric series to quickly calculate sums such as
sin(1°) + sin(2°) + … + sin(179°). Try it!
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