Answer:
[tex]\bold{4\sqrt3 + i4}[/tex]
Step-by-step explanation:
Given complex number is:
[tex][2(cos10^\circ + i sin10^\circ)]^3[/tex]
To find:
Answer in rectangular form after using De Moivre's theorem = ?
i.e. the form [tex]a+ib[/tex] (not in forms of angles)
Solution:
De Moivre's theorem provides us a way of solving the powers of complex numbers written in polar form.
As per De Moivre's theorem:
[tex](cos\theta+isin\theta)^n = cos(n\theta)+i(sin(n\theta))[/tex]
So, the given complex number can be written as:
[tex][2(cos10^\circ + i sin10^\circ)]^3\\\Rightarrow 2^3 \times (cos10^\circ + i sin10^\circ)^3[/tex]
Now, using De Moivre's theorem:
[tex]\Rightarrow 2^3 \times (cos10^\circ + i sin10^\circ)^3\\\Rightarrow 8 \times [cos(3 \times10)^\circ + i sin(3 \times10^\circ)]\\\Rightarrow 8 \times (cos30^\circ + i sin30^\circ)\\\Rightarrow 8 \times (\dfrac{\sqrt3}2 + i \dfrac{1}{2})\\\Rightarrow \dfrac{\sqrt3}2\times 8 + i \dfrac{1}{2}\times 8\\\Rightarrow \bold{4\sqrt3 + i4}[/tex]
So, the answer in rectangular form is:
[tex]\bold{4\sqrt3 + i4}[/tex]
