What is the general formula for finding the nth roots of z = r(cosθ + isinθ) using De Moivre's Theorem?
Understand the Problem
The question asks for the general formula to find the nth roots of a complex number z given in polar form, using De Moivre's Theorem. We need to identify the correct formula from the given options.
Answer
$z_k = \sqrt[n]{r} \left( \cos\left(\frac{\theta + 2\pi k}{n}\right) + i\sin\left(\frac{\theta + 2\pi k}{n}\right) \right)$
Answer for screen readers
The general formula to find the nth roots of a complex number z is: $$z_k = \sqrt[n]{r} \left( \cos\left(\frac{\theta + 2\pi k}{n}\right) + i\sin\left(\frac{\theta + 2\pi k}{n}\right) \right)$$
Steps to Solve
- Recall De Moivre's Theorem for roots
De Moivre's Theorem states that for a complex number $z = r(\cos(\theta) + i\sin(\theta))$, the $n$th roots are given by: $$z_k = \sqrt[n]{r} \left( \cos\left(\frac{\theta + 2\pi k}{n}\right) + i\sin\left(\frac{\theta + 2\pi k}{n}\right) \right)$$ where $k = 0, 1, 2, ..., n-1$.
- Identify the correct formula
Based on De Moivre's Theorem, compare to the provided options and identify the formula that correctly represents the $n$th roots of a complex number.
The general formula to find the nth roots of a complex number z is: $$z_k = \sqrt[n]{r} \left( \cos\left(\frac{\theta + 2\pi k}{n}\right) + i\sin\left(\frac{\theta + 2\pi k}{n}\right) \right)$$
More Information
De Moivre's Theorem is a powerful tool in complex number theory, allowing us to find powers and roots of complex numbers expressed in polar form. The formula for the $n$th roots provides $n$ distinct roots, equally spaced around a circle in the complex plane.
Tips
A common mistake is forgetting the $2\pi k$ term in the argument of the cosine and sine functions. This term is essential for finding all $n$ distinct roots. Also, confusing $n$ and $k$ is common, where $n$ represents the root you are finding (square root, cube root, etc.) and $k$ is the index of the root.
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