Given w= √ 2(cos(pi/4)+isin(pi/4)) and z=2(cos (pi/2)+i sin (pi/2)) what is w-z expressed in polar form?

"The polar form of complex number z = x + iy is [tex]z=r~(cos\theta+i~sin\theta)[/tex] where [tex]r=\sqrt{x^{2} +y^{2} }[/tex] and [tex]\theta=tan^{-1}(\frac{y}{x} )[/tex]"

For given question,

We have been given two complex numbers in polar form.

[tex]w=\sqrt{2}(cos(\frac{\pi}{4} )+i~sin(\frac{\pi}{4} ))[/tex] and [tex]z=2(cos(\frac{\pi}{2} )+i~sin(\frac{\pi}{2} ))[/tex]

We simplify above complex numbers.

[tex]w=\sqrt{2}(cos(\frac{\pi}{4} )+i~sin(\frac{\pi}{4} )) \\\\w=\sqrt{2}(\frac{1}{\sqrt{2} } +i~ \frac{1}{\sqrt{2} } ) \\\\w=1+i[/tex]

And the second complex number is,

[tex]z=2(cos(\frac{\pi}{2} )+i~sin(\frac{\pi}{2} ))\\\\z=2(0+i~1)\\\\z=0+2i[/tex]

Now if we find the value of w - z

⇒ w - z = (1 + i) - (0 + 2i)

⇒ w - z = 1 - i

Now we find the polar form of complex number.

[tex]r=\sqrt{1^{2} +(-1)^{2} } \\\\r=\sqrt{2}[/tex]

And the value of [tex]\theta[/tex] would be,

[tex]\theta=tan^{-1}(\frac{-1}{1} )\\\\\theta=tan^{-1}(-1)\\\\\theta=\frac{3\pi}{4}[/tex]

So, w - z in the polar form would be [tex]1+i=\sqrt{2}(cos(\frac{3\pi}{4} )+i~sin(\frac{3\pi}{4} ))[/tex]

Therefore, [tex]w -z =\sqrt{2}(cos(\frac{3\pi}{4} )+i~sin(\frac{3\pi}{4} ))[/tex]

Learn more about the complex numbers here:

https://brainly.com/question/17162901

#SPJ3