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The De Moivre's formula is:

$$ [r(cos\theta + (\imath *sin\theta ))]^n=r^n((cos * n\theta) + (i * sin * n\theta)) $$

The following two terms are complex numbers, which are the combinations of a Real Number and Imaginary Number:

$$ \imath *sin\theta\ $$

$$ i * sin * n\theta $$

• A Real Number is the type of number: 1.4, 5/8, -2390, 0, for example.
• An Imaginary Number gives a negative result when squared: i^2=-1

The complex number is

$$ 4 + 3\imath $$

r is

$$ r = \sqrt{4^2+3^2}=\sqrt{25}=5 $$

angle (in radian) is

$$ \Theta =tan^{-1}(y/x)=tan^{-1}(3/4)=0.6435 $$

x is

$$ cos(\theta)=x/r $$

$$ x=r*cos(\theta)=5*cos(0.6435)=4 $$

y is

$$ sin(\theta)=y/r $$

$$ y=r*sin(\theta)=5*sin(0.6435)=3 $$

Here is the common way to write the complex number below:

$$ x+(i*y)=r(cos\theta + (i*(sin\theta)))=r*cis\theta $$

• Note that a combination of cos and sin is often shortened to 'cis'

In the case, therefore, the complex number can be written as follows:

$$ 4+3i=5*cis(0.6435) $$

In the De Moivre's formula,

$$ [r(cos\theta + (\imath *sin\theta ))]^n=r^n((cos * n\theta) + (i * sin * n\theta)) $$

magnitude becomes

$$ r^n $$

angle (in radian) becomes

$$ n\theta $$

In the above case, the De Moivre formula is

$$ (5*cis(0.6435))^2=5^2*cis(2*0.6435)=25*cis(1.287) $$

So, the magnitude is 25 and the angle is 1.287 in radian.