Question

How it come that these equations below... This is De Moivre's Theorem , any proofs...

Answer 1

any help ...

Answer 2

The explanations of multiplication and division of complex numbers in polar form do not involve De Moivre's Formula. The explanations begin with the definition of multiplication, as follows:
\[\large z _{1}z _{2}=(x _{1}, y _{1})(x _{2}, y _{2})=(x _{1},x _{2}-y _{1} y _{2},\ \ x _{1} y _{2}+x _{2}y _{1})\]

Answer 3

Let \[\large z _{1}=r _{1}(\cos \theta _{1}+i \sin\theta _{1})\]
and
\[\large z _{2}=r _{2}(\cos \theta _{2}+i \sin \theta _{2})\]
then by the definition of multiplication the product is at first
\[\large z _{1}z _{2}=r _{1}r _{2}[(\cos \theta _{1}\cos \theta _{2}-\sin\ \theta _{1}\sin \theta _{2})+\]
\[\large i(\sin \theta _{1}\cos \theta _{2}+\cos \theta _{1}\sin \theta _{2})]\]

Answer 4

The addition rules for sine and cosine now give us:
\[\large z _{1}z _{2}=r _{1}r _{2}[\cos(\theta _{1}+\theta _{2})+i \sin(\theta _{1}+\theta _{2})]\]