Answer:
2.433
Step-by-step explanation:
using
a+bi=r(cosP+isinP)
Let
-5 = [tex]r_{1}[/tex]cosФ
12 = [tex]r_{1}[/tex]sinФ
Taking square both on both sides
(-5)² + (12)² = [tex]r_{1}^2[/tex]cos²Ф +[tex]r_{1}^2[/tex]sin²Ф
we know that
sin²Ф + cos²Ф = 1
25 + 144 = [tex]r_{1}^2[/tex](1)
[tex]r_{1}[/tex] = √(144 + 25)
= 13
Similarly,
Let
-3 = [tex]r_{2}[/tex]cosФ
-3√3 = [tex]r_{2}[/tex]sinФ
taking square both on both sides
(-3)² + (-3√3)² = [tex]r_{2}^2[/tex]cos²Ф +[tex]r_{2}^2[/tex]sin²Ф
we know that
sin²Ф + cos²Ф = 1
9 + 27 = [tex]r_{2}^2[/tex](2)
[tex]r_{2}[/tex] = √(9 + 27)
= 6
Also
[tex]\frac{r_{1}sin\alpha }{r_{1}cos\alpha } = \frac{12}{-5}[/tex]
[tex]\alpha = r_{1}tan^{-1} (\frac{12}{-5} )[/tex]
And
[tex]\frac{r_{2}sin\alpha }{r_{2}cos\alpha } = \frac{-3\sqrt{3} }{3}\\[/tex]
[tex]\alpha = r_{2}tan^{-1} \frac{-3\sqrt{3} }{3}[/tex]
So
[tex]\frac{-5+12i}{3-3\sqrt{3i} }[/tex] = [tex]\frac{13tan^{-1}\frac{-12}{5}}{6tan^{-1}(-\sqrt{3})}[/tex]
= 2.433
