In Exercises 65–68, find all the complex roots. Write roots in polar form with θ in degrees. The complex square roots of 25(cos 210° + i sin 210°)

Verified step by step guidance

1. Step 1: Express the complex number in polar form.
2. Step 2: Use De Moivre's Theorem to find the square roots.
3. Step 3: Calculate the magnitude of the square roots.
4. Step 4: Determine the angles for the square roots using the formula ${\theta }_k=\frac{\theta +360k}{n}$ where $n$ is the root degree and $k$ is an integer.
5. Step 5: Write the square roots in polar form as $r(\cos \theta +i\sin \theta )$.

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 their roots and performing operations involving them.

Polar Form of Complex Numbers

The polar form of a complex number expresses it in terms of its magnitude (r) and angle (θ), using the formula r(cos θ + i sin θ). This representation is particularly useful for multiplication, division, and finding roots of complex numbers. The angle θ is measured in degrees or radians and indicates the direction of the vector in the complex plane.

Finding Roots of Complex Numbers

To find the roots of a complex number in polar form, we use De Moivre's Theorem, which states that for a complex number r(cos θ + i sin θ), the nth roots can be found using the formula: r^(1/n)(cos(θ/n + k(360°/n)) + i sin(θ/n + k(360°/n))), where k is an integer from 0 to n-1. This method allows us to determine all distinct roots by varying k.