The expression that is closest to the polar form of z + w is
[tex]5(cos(143^{\circ})+i~sin(143^{\circ}))[/tex]
"The number of the form a + ib where a, b are real numbers and [tex]i=\sqrt{-1}[/tex]"
"The polar form of the complex number z = x + iy is [tex]z=r(cos(\theta)+i~sin(\theta))[/tex] where [tex]r=\sqrt{x^{2} +y^{2} } , ~\theta=tan^{-1}(\frac{y}{x} )[/tex] "
For given question,
We have been given two complex numbers.
[tex]z = 3\sqrt{2} (cos(135^{\circ}) + i~sin(135^{\circ})[/tex] and [tex]w = cos(180^{\circ}) + i~sin(180^{\circ})[/tex]
We know that,
[tex]cos(135^{\circ})\\=cos(90^{\circ}+45^{\circ})\\=-sin(45^{\circ})\\=-\frac{1}{\sqrt{2} }[/tex]
And
[tex]sin(135^{\circ})\\=sin(90^{\circ}+45^{\circ})\\=cos(45^{\circ})\\=\frac{1}{\sqrt{2} }[/tex]
So, the first complex number would be,
[tex]z = 3\sqrt{2} (cos(135^{\circ}) + i~sin(135^{\circ})\\\\z = 3\sqrt{2} (-\frac{1}{\sqrt{2} } + i~\frac{1}{\sqrt{2} } )\\\\z=-3+3i[/tex]
We know that [tex]cos(180^{\circ})=-1,~~sin(180^{\circ})=0[/tex]
So the second complex number would be,
[tex]w = cos(180^{\circ}) + i~sin(180^{\circ})\\\\w = -1 + 0i[/tex]
So, the sum of complex numbers z and w would be,
[tex]z+w\\=(-3+3i)+(-1+0i)\\=-4+3i[/tex]
We write the complex number -4 + 3i in the polar form.
Comparing above number with x + iy, we have x = -4 and y = 3
[tex]\Rightarrow r=\sqrt{(-4)^2+3^2}\\ \Rightarrow r=5[/tex]
Also, the value of [tex]\theta[/tex] would be,
[tex]\theta=tan^{-1}(\frac{3}{-4} )\\\\\theta= -36.86^{\circ}\\\\\theta=180^{\circ}-36.86^{\circ}\\\\\theta=143.14^{\circ}[/tex]
So the complex number -4 + 3i in the polar form would be, [tex]5(cos(143.14^{\circ})+i~sin(143.14^{\circ}))[/tex]
Therefore, the expression that is closest to the polar form of z + w is
[tex]5(cos(143^{\circ})+i~sin(143^{\circ}))[/tex]
Learn more about polar form of complex number here:
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