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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e^πi = cos(x) + isin(x) can be used to present trigonometric functions as complex exponential functions.
sin(x) = ( e^(ix) – e^(-ix) ) / 2i
cos(x) = ( e^(ix) + e^(-ix) ) / 2
At first these may seem overly complex (heh). But when you have an equation that contains a combination of sin and cos functions being added and multiplied together and you want to simplify the mess they become very useful.
How do you even simplify trigonometric functions? They are so often treated as mysterious black boxes where you put numbers in and get other numbers out.
Sometimes you may be able to come up with some geometric reasoning for the simplification. But often the only practical way to do it is to use the complex exponents. (the other option is usually to use infinite series and hope that the sum/multiplication of those infinite series converges nicely)
Lets for example say that you have sin(x)*cos(x) in some formula that you want to simplify.
sin(x)*cos(x)
Substitute with complex exponentials.
= ( e^(ix) – e^(-ix) ) / 2i * ( e^(ix) + e^(-ix) ) / 2
Do the multiplication
= ( e^2ix + e^0 -e^0 -e^-2ix ) / 4i
= (1/2) * ( e^i(2x) – e^-i(2x) ) / 2i
Notice that this here is the complex exponent form of cosine
= (1/2) * cos( 2x )
Now we know that sin(x)*cos(x) = 0.5*cos(2x). Isn’t that nice.
This way you can simplify all sort of nasty trigonometric function containing statements. For example:
sqrt( 1-sin^(2)(x) ) = cos(x)
( 1 – 4 * sin^(2)(x) ) * cos(x) = cos(3x)
2 * cos( x/2 + y/2 ) * sin( x/2 -y/2 ) = sin(x) – sin(y)
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