If-z-1-6-cos-pi-4-sin-pi-4-and-z-2-2-cos-pi-5-i-sin-pi-5-calculate-z-1-z-2-

Question Number 72787 by Maclaurin Stickker last updated on 02/Nov/19

I f z_{1} = 6 (\cos \frac{π}{4} + \sin \frac{π}{4}) a n d

z_{2} = 2 (\cos \frac{π}{5} + i \times \sin \frac{π}{5}) c a l c u l a t e \frac{z_{1}}{z_{2}} .

Commented by MJS last updated on 02/Nov/19

z_{1} \in R ? ? ?

Answered by MJS last updated on 02/Nov/19

z_{1} = 6 (\cos \frac{π}{4} + \sin \frac{π}{4}) = 6 \sqrt{2}

z_{2} = 2 (\cos \frac{π}{5} + i \times \sin \frac{π}{5}) = {2 e}^{i \frac{π}{5}} = \frac{1 + \sqrt{5}}{2} + \frac{\sqrt{10 - 2 \sqrt{5}}}{2} i

\Rightarrow \frac{z_{1}}{z_{2}} = 6 \sqrt{2} \times \frac{1}{2} e^{- i \frac{π}{5}} = 3 \sqrt{2} e^{- i \frac{π}{5}} = 3 \sqrt{2} (\cos \frac{π}{5} - i \times \sin \frac{π}{5}) =

= \frac{3 \sqrt{2} (1 + \sqrt{5})}{4} - \frac{3 \sqrt{5 - \sqrt{5}}}{2} i

or

z_{1} = 6 (\cos \frac{π}{4} + i \times \sin \frac{π}{4}) = {6 e}^{i \frac{π}{4}} = 3 \sqrt{2} + 3 \sqrt{2} i

z_{2} = 2 (\cos \frac{π}{5} + i \times \sin \frac{π}{5}) = {2 e}^{i \frac{π}{5}} = \frac{1 + \sqrt{5}}{2} + \frac{\sqrt{10 - 2 \sqrt{5}}}{2} i

\Rightarrow \frac{z_{1}}{z_{2}} = {6 e}^{i \frac{π}{4}} \times \frac{1}{2} e^{- i \frac{π}{5}} = {3 e}^{i \frac{π}{20}} = 3 (\cos \frac{π}{20} + \sin \frac{π}{20}) =

= \frac{3 (\sqrt{2} (1 + \sqrt{5}) + 2 \sqrt{5 - \sqrt{5}})}{8} + \frac{3 (\sqrt{2} (1 + \sqrt{5}) - 2 \sqrt{5 - \sqrt{5}})}{8} i

Commented by Maclaurin Stickker last updated on 03/Nov/19

T h a n k y o u, s i r M J S .