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Tresset [83]
2 years ago
11

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

Mathematics
1 answer:
slava [35]2 years ago
3 0

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}

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