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

After solving the power:

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

Rectangular form:

`\bold{1+i\sqrt3}`

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 * 20)^\circ+isin(3 * 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 = (1)/(2)` and
`sin60^\circ = (\sqrt3)/(2)`

`2((1)/(2)+i\frac{\sqrt3}2)\\\Rightarrow (2 * (1)/(2)+i\frac{\sqrt3}2* 2)\\\Rightarrow \bold{1 +i\sqrt3 }`

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

`\bold{1+i\sqrt3}`