Let‘s explore some more mysterious consequences of Euler formula,
eίθ = cos(θ) + ί sin(θ) ———————-(1)
I’m gonna do some detailed maths on this. But, the result we get is quite interesting.
By replacing θ by -θ we get,
e–ίθ = cos(-θ) + ί sin(-θ)
Then we have ,
e–ίθ = cos(θ) – ί sin(θ) ———————-(2)
From (1) and (2) we get,
Similarly , for some α ,
Suppose cos(θ) = cos(α) then ,
The above equation can be treated as a quadratic equation of eίθ whose solutions are ,
eία and e–ία .
When eίθ=eία ,
In general ίθ=ία + 2nπί for some integer n (why? Refer my previous article 😉 )
Therefore θ=2nπ + α
Proceding in a simillar manner , from the other root ( e–ία ) we can show that ,
θ=2nπ– α
So , our conclusion is that when cos(θ) = cos(α) we have θ=2nπ ± α.
I’m sure that most of you are familiar with this result.This is something you learn in your high school.But I’m not sure whether you have learnt the proof.But , if you have I’m sure , that’s using a classical argument by analysing the graph of y=cos(x) . But , here I’ve given a much different approach using Euler’s formula which involves the well known ghostly constant “e” and complex numbers.When we think about the origin , basically cosines came from GEOMETRY .”e” came from CALCULUS . And complex numbers came out of NOWHERE.!!!.
The essence of what I’m trying to say is in mathematics, things are interconnected in a mysterious way.Some of those beauties are discovered now.We know a lot more than ancient Greeks.But , a lot more to be discovered.Mathematics is not just something which can be used as a tool in other sciences. It’s all about the ultimate truth……
Image source : http://www.theguardian.com