In Exercises 65–68, find all the complex roots. Write roots in polar form with θ in degrees. The complex cube roots of 8(cos 210° + i sin 210°)

Verified step by step guidance

1. <insert step 1> Convert the given complex number into polar form. The given complex number is already in polar form as 8(cos 210° + i sin 210°), where the modulus is 8 and the argument is 210°.
2. <insert step 2> Use De Moivre's Theorem to find the cube roots. According to De Moivre's Theorem, the nth roots of a complex number with modulus r and argument θ are given by: r^{1/n} (cos((θ + 360°k)/n) + i sin((θ + 360°k)/n)), where k = 0, 1, ..., n-1.
3. <insert step 3> Calculate the modulus of the cube roots. Since the original modulus is 8, the modulus of each cube root is 8^{1/3}.
4. <insert step 4> Calculate the arguments for each of the three cube roots. Substitute n = 3 and k = 0, 1, 2 into the formula: (210° + 360°k)/3.
5. <insert step 5> Write each of the three cube roots in polar form using the calculated modulus and arguments.>

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'a' is the real part and 'b' is the imaginary part. In trigonometric form, a complex number can be represented as r(cos θ + i sin θ), where r is the modulus and θ is the argument. Understanding complex numbers is essential for finding roots and performing operations in the complex plane.

De Moivre's Theorem

De Moivre's Theorem states that for any complex number in polar form r(cos θ + i sin θ) and any integer n, the nth power of the complex number can be expressed as r^n(cos(nθ) + i sin(nθ)). This theorem is also used to find the nth roots of complex numbers, which involves dividing the angle by n and taking the nth root of the modulus, making it crucial for solving the given problem.

Polar Coordinates

Polar coordinates represent points in the plane using a distance from the origin (r) and an angle (θ) from the positive x-axis. In the context of complex numbers, this representation simplifies multiplication and division, as well as finding roots. When converting complex numbers to polar form, understanding how to manipulate angles and magnitudes is key to accurately determining the roots.