Find all complex number solutions of each equation. Write answers in trigonometric form. See Exa

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Find all complex number solutions of each equation. Write...

Key Concepts

Complex Numbers in Trigonometric Form

This concept involves expressing complex numbers in the form r(cos ? + i sin ?), where r represents the modulus (or absolute value) of the complex number and ? represents its argument (or angle with the positive real axis). This representation simplifies multiplication, division, and computation of powers and roots of complex numbers.

Polar Representation of Complex Numbers

The polar form of a complex number is a way to represent it using its distance from the origin (modulus) and the angle it makes with the positive real axis (argument). This representation is particularly useful when performing operations like exponentiation and extraction of roots, as it leverages the periodic nature of trigonometric functions.

De Moivre's Theorem

De Moivre's Theorem states that for any complex number expressed in trigonometric form, its powers can be calculated by raising the modulus to the power and multiplying the angle by that power. Specifically, (r(cos ? + i sin ?))^n = r^n (cos(n?) + i sin(n?)). This theorem is fundamental for finding the nth roots of complex numbers.

Finding nth Roots of Complex Numbers

Finding the nth roots of a complex number involves determining all distinct solutions to the equation x^n = r(cos ? + i sin ?). This is done by taking the nth root of the modulus and distributing the argument evenly with an added factor of 2?k (where k is an integer) to account for the periodicity of the trigonometric functions. The general solution is given by x = r^(1/n) [cos((? + 2?k)/n) + i sin((? + 2?k)/n)], with different values of k providing all distinct roots.

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