Wednesday, 3 September 2014

complex numbers - Why: z1=cos(phi)+isin(phi)



Let's suppose that we have a complex number with




r=1z=cosϕ+isinϕ



Then why is z1=cos(ϕ)+isin(ϕ)=cosϕisinϕ


Answer




This is De Moivre's formula
in action:
For complex z and any integer n we have
zn=(cos(φ)+isin(φ))n=cos(nφ)+isin(nφ)




The formula
z1=cos(φ)+isin(φ)
is De Moivre's formula evaluated at n=1.


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