Respuesta :
Part I: The complex number √3-i in the polar form will be [tex]2(cos210^0+isin210^0)[/tex]
The complex number -2-2i in the polar form will be [tex]2\sqrt{2}(cos45^0+isin45^0)[/tex]
Part II: The expression of the product of wz in polar form is [tex]4\sqrt{2-\sqrt{3} } (cos45^0+isin45^0)[/tex]
Part III: The expression of z² value in polar form is [tex]z^2 = 8(cos\frac{\pi}{8 }+isin \frac{\pi}{8} )\\[/tex]
Part IV: The expression of w^4 in polar form is expressed as [tex]16(cos52.5^0+isin52.5^0)[/tex]
Complex numbers are the square roots of any negative numbers. They have both real and imaginary axis.
The rectangular form of expressing complex numbers is expressed as:
- z = x + iy
The polar representation is expressed as:
- r(cosθ - isinθ)
Given the complex numbers
z=√3-i
w=-2-2i
Part I: To express in polar form, we need to first get the modulus and argument of each of the complex numbers.
z= √3-i
|z| = [tex]\sqrt{(\sqrt{3} )^2+(-1)^2} \\[/tex]
[tex]|z| = \sqrt{3+1} \\|z|=\sqrt{4}\\|z|=2[/tex]
For the argument:
[tex]\theta = tan^{-1}\frac{y}{x}\\ \theta = tan^{-1}\frac{-1}{\sqrt{3} }\\\theta =-30^0\\[/tex]
Since tan is negative in the 3rd quadrant
[tex]\theta = 180 + 30\\\theta = 210^0[/tex]
The complex number √3-i in the polar form will be [tex]2(cos210^0+isin210^0)[/tex]
For the complex number:
z= -2-2i
|z| = [tex]\sqrt{(-2 )^2+(-2)^2} \\[/tex]
[tex]|z| = \sqrt{4+4} \\|z|=\sqrt{8}\\|z|=2\sqrt{2}[/tex]
For the argument:
[tex]\theta = tan^{-1}\frac{y}{x}\\ \theta = tan^{-1}\frac{-2}{\sqrt{-2} }\\\theta=tan^{-1}1\\\theta =45^0\\[/tex]
The complex number -2-2i in the polar form will be [tex]2\sqrt{2}(cos45^0+isin45^0)[/tex]
Part II: Taking the product of wz
(√3 - 1)(-2-2i)
Expand
wz = -2√3-2√3 i + 2 + 2i
wz = (-2√3+2) +(-2√3+2)i
Get the modulus
[tex]=\sqrt{(-2\sqrt{3}+2)^2+(-2\sqrt{3}+2)^2} \\=\sqrt{12-8\sqrt{3}+4 + 12-8\sqrt{3}+4 } \\=\sqrt{24-16\sqrt{3}+8}\\=\sqrt{32-16\sqrt{3} }\\=\sqrt{16(2-\sqrt{3} )}\\=4\sqrt{2-\sqrt{3} }[/tex]
Get the argument:
[tex]\theta = tan^{-1}\frac{-2\sqrt{3}+2}{-2\sqrt{3}+2}\\ \theta = tan^{-1}1\\\theta =45^0[/tex]
Expression in polar form is [tex]4\sqrt{2-\sqrt{3} } (cos45^0+isin45^0)[/tex]
Part III: To calculate z², we will simply square the polar form of z and apply De Moivre's theorem as shown:
Since z = [tex]2\sqrt{2}(cos45^0+isin45^0)[/tex]
[tex]z^2 = [2\sqrt{2}(cos\frac{\pi}{4}+isin\frac{\pi}{4} )]^2\\z^2 = (2\sqrt{2})^2(cos\frac{1}{2} * \frac{\pi}{4 }+isin\frac{1}{2} * \frac{\pi}{4} )]\\z^2 = 8(cos\frac{\pi}{8 }+isin \frac{\pi}{8} )\\[/tex]
This shows that the expression of z² value in polar form is [tex]z^2 = 8(cos\frac{\pi}{8 }+isin \frac{\pi}{8} )\\[/tex]
Part 4: We will also use De Moivre's theorem to get w⁴
Since w= [tex]2(cos210^0+isin210^0)[/tex]
[tex]w^4 = [2(cos210+isin210)]^4\\w^4 = 2^4(cos\frac{210}{4} +isin\frac{210}{4} )\\w^4=16(cos52.5^0+isin52.5^0)[/tex]
Hence the expression of w^4 in polar form is expressed as [tex]16(cos52.5^0+isin52.5^0)[/tex]
Learn more about complex numbers here: https://brainly.com/question/12375854
