Question

Use​ DeMoivre's Theorem to find the indicated power of the complex number. Write the answer in rectangular form.

2(cos20∘+isin20∘))3=__________

Answer 1

Answer:

After solving the power:

$\bold{2(cos60^\circ+isin60^\circ)}$

Rectangular form:

$\bold{1+i\sqrt3}$

Step-by-step explanation:

Given the complex number:

$2(cos20^\circ+isin20^\circ)^3$

To find:

The indicated power by using De Moivre's theorem.

The complex number in rectangular form.

Rectangular form of a complex number is given as $a+ib$ where a and b are real numbers.

Solution:

First of all, let us have a look at the De Moivre's theorem:

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

First of all, let us solve:

$(cos20^\circ+isin20^\circ)^3$

Let us apply the De Moivre's Theorem:

Here, n = 3

$(cos20^\circ+isin20^\circ)^3 = cos(3 \times 20)^\circ+isin(3 \times 20)^\circ\\\Rightarrow cos60^\circ+isin60^\circ$

Now, the given complex number becomes:

$2(cos60^\circ+isin60^\circ)$

Let us put the values of $cos60^\circ = \frac{1}{2}$ and $sin60^\circ = \frac{\sqrt3}{2}$

$2(\dfrac{1}{2}+i\dfrac{\sqrt3}2)\\\Rightarrow (2 \times \dfrac{1}{2}+i\dfrac{\sqrt3}2\times 2)\\\Rightarrow \bold{1 +i\sqrt3 }$

So, the rectangular form of the given complex number is:

$\bold{1+i\sqrt3}$