Chapter 8, Problem 7RCC

(a)

To determine

To state: De Moivre’s Theorem.

Answer to Problem 7RCC

De Moivre’s Theorem was stated.

Explanation of Solution

Given:

Calculation:

De Moivre's theorem:

De Moivre's theorem states that to take the nth power of a complex number, take the nth power of the modulus and multiply the argument by n .

If $z=r\left(\cos \theta +i\sin \theta \right)$ , then for any integer n

  $z^n=r^n\left(\cos n\theta +i\sin n\theta \right)$.

Conclusion:

De Moivre’s Theorem was stated.

(b)

To determine

To find: the nth roots of a complex number.

Answer to Problem 7RCC

The nth roots of a complex number is $W_k=r^{\frac{1}{n}}\left[\cos \left(\frac{\theta +2k\pi }{n}\right)+i\sin \left(\frac{\theta +2k\pi }{n}\right)\right]\text{, for }k=0,1,2,\mathrm{...},n-1.$

Explanation of Solution

Calculation:

the procedure to find the nth roots of a complex number.

The nth roots of complex number $z=r\left(\cos \theta +i\sin \theta \right)$ when 'n' is a positive integer is

  $W_k=r^{\frac{1}{n}}\left[\cos \left(\frac{\theta +2k\pi }{n}\right)+i\sin \left(\frac{\theta +2k\pi }{n}\right)\right]\text{, for }k=0,1,2,\mathrm{...},n-1.$

For finding the nth roots of $z=r(\cos \theta +i\sin \theta )$ , perform the following steps

1. The modulus of each nth root is $r^{\frac{1}{n}}$ .

2. The argument of the first root is $\frac{\theta }{n}$ .

3. repeatedly add $\frac{2\pi }{n}$ to get the argument of each successive root.

Conclusion:

Therefore, the nth roots of a complex number is $W_k=r^{\frac{1}{n}}\left[\cos \left(\frac{\theta +2k\pi }{n}\right)+i\sin \left(\frac{\theta +2k\pi }{n}\right)\right]\text{, for }k=0,1,2,\mathrm{...},n-1.$