Monday, 17 July 2017

algebra precalculus - Imaginary cos1 value significance?

When I was bored in AP Psych last year, I jokingly asked myself if there was a cosine inverse of 2. Curious about it, I tried calculating it as follows:
cos(x)=2sin(x)=1cos2(x)=14=±i3
Then, by Euler's formula, you have
eix=cos(x)+isin(x)eix=2±3ix=ln(2±3)x=iln(2±3)



So, there was a way to calculate the inverse cosine of numbers whose magnitude is greater than 1 (this was verified on Wolfram Alpha). To what extent is this kind of calculation valid? Does it have any interesting applications/implications in math, or any other subjects? Thanks. :)



Edit I just realized this is very easily explained by 2cos(x)=eix+eix, but I'm still curious if this has any significance/intuition.

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