Answer:
[tex]w-z=\sqrt{2}\biggr(cos(\frac{3\pi}{4})+isin(\frac{3\pi}{4})\biggr)[/tex]
Step-by-step explanation:
Convert from polar to rectangular
[tex]w=\sqrt{2}(cos(\frac{\pi}{4})+isin(\frac{\pi}{4}))\\\\w=\sqrt{2}(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i)\\ \\w=1+i[/tex]
[tex]z=2(cos(\frac{\pi}{2})+isin(\frac{\pi}{2}))\\ \\z=2(0+i)\\\\z=2i[/tex]
Subtract the complex numbers
[tex]w-z=(1+i)-2i=1+i-2i=1-i[/tex]
Convert from rectangular (x,y) to polar (r,θ)
[tex]r=\sqrt{x^2+y^2}\\r=\sqrt{1^2+(-1)^2}\\r=\sqrt{1+1}\\r=\sqrt{2}\\\\\theta=tan^{-1}(\frac{y}{x})\\ \theta=tan^{-1}(\frac{-1}{1})\\ \theta=tan^{-1}(-1)\\\theta=\frac{3\pi}{4}\\ \\w-z=\sqrt{2}\biggr(cos(\frac{3\pi}{4})+isin(\frac{3\pi}{4})\biggr)[/tex]
