Interesting Relationship

As n becomes larger Cos (2π/n) → 1.

So for example when n = 1000, Cos (2π/n) = .99998026...

Also when n becomes larger Sin (2π/n) → 2π/n.

Thus again when n = 1000, Sin (2π/n) = .0062831439... = (6.2831439.../1000)

(and 2π = 6.2831853...),

and with respect to Euler's Identity,

e^(iπ) = - 1;

therefore

e^(2iπ) = 1 (i.e. 1^1)


Also,

e^(2iπ) = Cos (2π) + i Sin (2π).

Therefore e^{(2iπ)/n} = Cos {(2π)/n} + i Sin {(2π)/n}.

So as n → ∞,

e^{(2iπ)/n} = 1^(1/n) → 1 + (2iπ)/n