Question

Use De Moivre's theorem to find the indicated power of the complex number. Write the answer in rectangular form.[2(cos10∘ + i sin10∘)]^3.

Answer 1

Answer:

$\bold{4\sqrt3 + i4}$

Step-by-step explanation:

Given complex number is:

$[2(cos10^\circ + i sin10^\circ)]^3$

To find:

Answer in rectangular form after using De Moivre's theorem = ?

i.e. the form $a+ib$ (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:

$(cos\theta+isin\theta)^n = cos(n\theta)+i(sin(n\theta))$

So, the given complex number can be written as:

$[2(cos10^\circ + i sin10^\circ)]^3\\\Rightarrow 2^3 \times (cos10^\circ + i sin10^\circ)^3$

Now, using De Moivre's theorem:

$\Rightarrow 2^3 \times (cos10^\circ + i sin10^\circ)^3\\\Rightarrow 8 \times [cos(3 \times10)^\circ + i sin(3 \times10^\circ)]\\\Rightarrow 8 \times (cos30^\circ + i sin30^\circ)\\\Rightarrow 8 \times (\dfrac{\sqrt3}2 + i \dfrac{1}{2})\\\Rightarrow \dfrac{\sqrt3}2\times 8 + i \dfrac{1}{2}\times 8\\\Rightarrow \bold{4\sqrt3 + i4}$

So, the answer in rectangular form is:

$\bold{4\sqrt3 + i4}$