How to Find the Cube Roots of a Complex Number Example with -1 + sqrt(3)*i | Summary and Q&A

TL;DR

This video explains the process of finding the cube roots of a complex number in trigonometric form.

Key Insights

• 💁 Complex numbers can be represented in trigonometric form using the modulus and argument.
• #️⃣ The modulus of a complex number can be calculated using the formula sqrt(x^2 + y^2).
• ⭕ The argument of a complex number can be determined by reasoning and referencing the unit circle.
• 🫚 The formula for finding the roots of a complex number includes the cube root of the modulus and the argument divided by the desired root (e.g., 3 for cube roots).
• 😉 The distinct roots can be obtained by varying the value of k in the formula.
• 😑 Writing complex numbers in trigonometric form allows for the expression of roots involving trigonometric functions.
• 🫚 The process of finding the roots of a complex number requires careful calculations and attention to detail.

Transcript

hi everyone in this problem we're going to find the cube roots of this complex number so in order to find the roots of a complex number the very first step is to write it in what's called trigonometric form or polar form so let's go ahead and do that first and then there's a formula we can use that will allow us to find the roots we're also going t... Read More

Questions & Answers

Q: What is the first step in finding the cube roots of a complex number?

The first step is to write the complex number in trigonometric form, using the formula r(cosθ + isinθ).

Q: How can the modulus of a complex number be found?

The modulus can be found using the formula sqrt(x^2 + y^2), where x and y are the real and imaginary parts of the complex number.

Q: How is the argument of a complex number determined?

The argument can be reasoned by referencing the unit circle and identifying the corresponding angle based on the real and imaginary parts.

Q: What is the formula for finding the roots of a complex number?

The formula is z_k = (n√r) * [cos(θ/n + 2kπ/n) + isin(θ/n + 2kπ/n)], where n is the desired root (e.g., 3 for cube roots) and k represents the distinct roots.

Summary & Key Takeaways

• The first step in finding the roots of a complex number is to write it in trigonometric form, using the formula r(cosθ + isinθ).

• The modulus (r) can be found using the formula sqrt(x^2 + y^2), where x and y are the real and imaginary parts of the complex number.

• The argument (θ) can be determined by reasoning and referencing the unit circle.

• The roots can be found using the formula z_k = (n√r) * [cos(θ/n + 2kπ/n) + isin(θ/n + 2kπ/n)], where n is the desired root (e.g., 3 for cube roots) and k represents the distinct roots.