Problem 17 - Contemporary Precalculus

Question

Find the nth roots in polar form. $$36\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right) ; \quad n=2$$

Step-by-Step Solution

Answer

Question: Find the square roots of the given complex number in polar form: $$36\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)$$. Answer: The square roots of the given complex number in polar form are: $$6\left(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}\right)$$ and $$6\left(\cos \frac{5\pi}{6} + i \sin \frac{5\pi}{6}\right)$$.

Step 1: Identify the given values

The polar form of the given complex number is: $$z = 36\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)$$ Here, the modulus $$r$$ is 36, the angle $$\theta$$ is $$\frac{\pi}{3}$$, and the number of roots $$n$$ is 2.

Step 2: Use De Moivre's theorem

Using De Moivre's theorem, the square roots of the complex number are given by: $$w_k = \sqrt[2]{36} \left( \cos \frac{\frac{\pi}{3}+2\pi k}{2} + i \sin \frac{\frac{\pi}{3}+2\pi k}{2} \right)$$, where $$k = 0, 1$$.

Step 3: Find the first square root (k=0)

For $$k = 0$$, we have: $$w_0 = \sqrt[2]{36} \left( \cos \frac{\frac{\pi}{3}}{2} + i \sin \frac{\frac{\pi}{3}}{2} \right)$$ $$w_0 = 6 \left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right)$$

Step 4: Find the second square root (k=1)

For $$k = 1$$, we have: $$w_1 = \sqrt[2]{36} \left( \cos \frac{\frac{\pi}{3}+2\pi}{2} + i \sin \frac{\frac{\pi}{3}+2\pi}{2} \right)$$ $$w_1 = 6 \left( \cos \frac{5\pi}{6} + i \sin \frac{5\pi}{6} \right)$$

Step 5: Write down the solution

So the square roots of the given complex number in polar form are: $$w_0 = 6\left(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}\right)$$ and $$w_1 = 6\left(\cos \frac{5\pi}{6} + i \sin \frac{5\pi}{6}\right)$$.

Key Concepts

De Moivre's Theorem

Understanding De Moivre's theorem is crucial when working with complex numbers, especially when it comes to calculating powers and roots. The theorem states that for any complex number written in polar form as \( z = r(\text{cos} \theta + i \text{sin} \theta) \) and any positive integer \( n \), the \( n \)th power of \( z \) is given by:
\[ z^n = \big[ r(\text{cos} \theta + i \text{sin} \theta) \big]^n = r^n(\text{cos}(n\theta) + i \text{sin}(n\theta)) \]
This formula is also pivotal when extracting roots of a complex number. To find the nth roots, we apply the theorem inversely, effectively distributing the fraction \(1/n\) to both the radius and the angle:
\[ z^{1/n} = r^{1/n}\big(\text{cos}(\frac{\theta + 2k\pi}{n}) + i \text{sin}(\frac{\theta + 2k\pi}{n})\big)\],
where \( k \) is an integer ranging from 0 to \(n-1\). Each value of \( k \) produces a distinct root, evenly spaced around the circle of radius \( r^{1/n} \) in the complex plane. This theorem essentially turns the challenging task of finding complex roots into a more manageable series of angles and radii calculations.

Complex Numbers

At the heart of operations like finding nth roots in polar form lies the concept of complex numbers. A complex number consists of a real part and an imaginary part and is expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with the property \( i^2 = -1 \).

Why are they important?

Complex numbers extend the real number system and allow for the solution of equations that have no real solutions, such as \( x^2 + 1 = 0 \). They also provide an expansive field in mathematics where operations such as addition, subtraction, multiplication, and division (except by zero) can be performed.
In polar form, a complex number takes advantage of trigonometry and the complex plane: \( z = r(\text{cos} \theta + i \text{sin} \theta) \) where \( r \) is the modulus (the distance from the origin to the point) and \( \theta \) is the argument (the angle from the positive real axis to the line connecting the origin to the point). Polar form is especially useful for multiplicative operations of complex numbers including the finding of roots, as it simplifies the process significantly.

Polar Coordinates

When we talk about nth roots in polar form, we delve into the realm of polar coordinates, which is an alternative to the Cartesian coordinate system most are familiar with.

Coordinate Conversion

In polar coordinates, a point is defined by its distance from the origin (radius, \( r \)) and the angle (\( \theta \)) it makes with the positive real axis. This system is particularly useful for circular and spiral shapes where points are more naturally positioned based on how far and at what angle they are relative to a central point.
In terms of our exercise with complex numbers, the polar coordinate system allows us to represent complex numbers as points in the complex plane. We can convert between Cartesian coordinates \( (x, y) \) and polar coordinates \( (r, \theta) \) using trigonometry:

• \( r = \sqrt{x^2 + y^2} \) (the modulus of the complex number)
• \( \theta = \text{atan2}(y,x) \) (the argument of the complex number)

Polar coordinates make multiplication and finding powers of complex numbers straightforward through De Moivre's theorem – which essentially boils down to multiplying the moduli and adding the angles.