## anonymous one year ago How it come that these equations below... This is De Moivre's Theorem , any proofs...

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1. anonymous

2. anonymous

3. anonymous

any help ...

4. kropot72

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: $\large z _{1}z _{2}=(x _{1}, y _{1})(x _{2}, y _{2})=(x _{1},x _{2}-y _{1} y _{2},\ \ x _{1} y _{2}+x _{2}y _{1})$

5. kropot72

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 $\large z _{1}z _{2}=r _{1}r _{2}[(\cos \theta _{1}\cos \theta _{2}-\sin\ \theta _{1}\sin \theta _{2})+$ $\large i(\sin \theta _{1}\cos \theta _{2}+\cos \theta _{1}\sin \theta _{2})]$

6. kropot72

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