Respuesta :
[tex]\bf r\left[ cos\left( \theta \right)+i\ sin\left( \theta \right) \right]\quad
\begin{cases}
x=rcos(\theta )\\
y=rsin(\theta )
\end{cases}\implies
\begin{array}{llll}
x&,&y\\
a&&b
\end{array}\implies a+bi\\\\
-------------------------------\\\\
2\left[ cos\left( 135^o\right)+i\ sin\left( 135^o\right) \right]\impliedby r=2\qquad \theta =135^o
\\\\\\
2\left( -\frac{\sqrt{2}}{2} \right)+i\ 2\left( \frac{\sqrt{2}}{2}\right)\implies -\sqrt{2}+\sqrt{2}\ i[/tex]
[tex]\bf -------------------------------\\\\ 3\left[ cos\left( 120^o\right)+i\ sin\left( 120^o\right) \right]\impliedby r=3\qquad \theta =120^o \\\\\\ 3\left( -\frac{1}{2} \right)+i\ 3\left( \frac{\sqrt{3}}{2}\right)\implies -\frac{3}{2}+\frac{3\sqrt{3}}{2}\ i[/tex]
[tex]\bf \\\\ -------------------------------\\\\ 5\left[ cos\left( \frac{5\pi }{4}\right)+i\ sin\left( \frac{5\pi }{4}\right) \right]\impliedby r=5\qquad \theta =\frac{5\pi }{4} \\\\\\ 5\left( -\frac{\sqrt{2}}{2} \right)+i\ 5\left( -\frac{\sqrt{2}}{2}\right)\implies -\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\ i[/tex]
[tex]\bf -------------------------------\\\\ 4\left[ cos\left( \frac{5\pi }{3}\right)+i\ sin\left( \frac{5\pi }{3}\right) \right]\impliedby r=4\qquad \theta =\frac{5\pi }{3} \\\\\\ 4\left( \frac{1}{2} \right)+i\ 4\left( -\frac{\sqrt{3}}{2}\right)\implies \frac{1}{2}-\frac{\sqrt{3}}{2}\ i[/tex]
[tex]\bf -------------------------------\\\\ 3\left[ cos\left( 120^o\right)+i\ sin\left( 120^o\right) \right]\impliedby r=3\qquad \theta =120^o \\\\\\ 3\left( -\frac{1}{2} \right)+i\ 3\left( \frac{\sqrt{3}}{2}\right)\implies -\frac{3}{2}+\frac{3\sqrt{3}}{2}\ i[/tex]
[tex]\bf \\\\ -------------------------------\\\\ 5\left[ cos\left( \frac{5\pi }{4}\right)+i\ sin\left( \frac{5\pi }{4}\right) \right]\impliedby r=5\qquad \theta =\frac{5\pi }{4} \\\\\\ 5\left( -\frac{\sqrt{2}}{2} \right)+i\ 5\left( -\frac{\sqrt{2}}{2}\right)\implies -\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}\ i[/tex]
[tex]\bf -------------------------------\\\\ 4\left[ cos\left( \frac{5\pi }{3}\right)+i\ sin\left( \frac{5\pi }{3}\right) \right]\impliedby r=4\qquad \theta =\frac{5\pi }{3} \\\\\\ 4\left( \frac{1}{2} \right)+i\ 4\left( -\frac{\sqrt{3}}{2}\right)\implies \frac{1}{2}-\frac{\sqrt{3}}{2}\ i[/tex]
The complex number 2(cos(135)+i sin(135)) in the polar form is -√2 + i√2 after calculating.
What is a complex number?
It is defined as the number which can be written as x+iy where x is the real number or real part of the complex number and y is the imaginary part of the complex number and i is the iota which is nothing but a square root of -1.
We know:
[tex]\rm Z= r(cos\theta + i \ sin\theta )[/tex] is the complex number in polar form.
a) 2(cos(135)+i sin(135))
Here r = 2
x = rcosθ
y = rsinθ
θ = 135 degree
By plugging and solving:
x = -√2
y = √2
Complex number in polar form:
= x + iy = -√2 + i√2
Similarly, we can change the rest of the complex number into the polar form.
b) 3(cos(120))+i sin(120)) = -3/2 +i 3√3/2
c) 5(cos(5pi/4)+i sin(5pi/4)) = -√2/2 -i √2/2
d) 4(cos(5pi/3)+i sin (5pi/3)) = 1/2 -i √3/2
Thus, the complex number 2(cos(135)+i sin(135)) in the polar form is -√2 + i√2 after calculating.
Learn more about the complex number here:
brainly.com/question/10251853
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