Problem 67 - College Trigonometry

Question

Find the indicated complex roots. Express your answers in polar form and then convert them into rectangular form. the two square roots of \(z=1+i \sqrt{3}\)

Step-by-Step Solution

Answer

The square roots of \(1+i\sqrt{3}\) are \(\frac{\sqrt{6}}{2} + i\frac{\sqrt{2}}{2} \) and \(-\frac{\sqrt{6}}{2} + i\frac{\sqrt{2}}{2}\).

Step 1: Express the Complex Number in Polar Form

First, identify the real and imaginary parts of the complex number. For \(z = 1 + i\sqrt{3}\), the real part is 1 and the imaginary part is \(\sqrt{3}\). To convert to polar form, calculate the modulus \(r\) using \(r = \sqrt{1^2 + (\sqrt{3})^2}\). This gives \(r = \sqrt{4} = 2\). Next, find the argument \(\theta\) using the formula \(\theta = \tan^{-1}\left(\frac{\sqrt{3}}{1}\right) = \frac{\pi}{3}\). Thus, the polar form of \(z\) is \(z = 2\left(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}\right)\).

Step 2: Find the Square Roots in Polar Form

To find the square roots of \(z\), use the formula for roots of complex numbers: \(z^{1/n} = r^{1/n}\left(\cos\left(\frac{\theta + 2k\pi}{n}\right) + i\sin\left(\frac{\theta + 2k\pi}{n}\right)\right)\), where \(n=2\), \(r=2\), and \(\theta = \frac{\pi}{3}\). Calculate the two roots for \(k = 0\) and \(k = 1\). For \(k=0\), the root is \(\sqrt{2}\left(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\right)\). For \(k=1\), the root is \(\sqrt{2}\left(\cos\frac{5\pi}{6} + i\sin\frac{5\pi}{6}\right)\).

Step 3: Convert the Roots to Rectangular Form

Convert each root from polar to rectangular form. For \(k=0\): \(\sqrt{2}(\cos(\frac{\pi}{6}) + i\sin(\frac{\pi}{6})) = \sqrt{2}(\frac{\sqrt{3}}{2} + i\frac{1}{2}) = \frac{\sqrt{6}}{2} + i\frac{\sqrt{2}}{2}\). For \(k=1\): \(\sqrt{2}(\cos(\frac{5\pi}{6}) + i\sin(\frac{5\pi}{6})) = \sqrt{2}(-\frac{\sqrt{3}}{2} + i\frac{1}{2}) = -\frac{\sqrt{6}}{2} + i\frac{\sqrt{2}}{2}\).

Key Concepts

Polar Form

Complex numbers can be represented in different ways, and one of the most intuitive methods is using the polar form. This form considers two main components:

• Modulus (r): The length of the vector which is the distance from the origin to the point in the complex plane. Calculate using the formula: \( r = \sqrt{a^2 + b^2} \), where \( a \) is the real part and \( b \) is the imaginary part of the number.
• Argument (\( \theta \)): The angle formed by the positive real axis and the line representing the complex number, calculated with: \( \theta = \tan^{-1}\left(\frac{b}{a}\right) \).

In polar form, a complex number \( z = a + bi \) can be expressed as:\[z = r(\cos \theta + i\sin \theta)\]This form offers a powerful way to simplify multiplication and division of complex numbers, as well as taking powers and roots, making calculations much easier. Polar form seamlessly integrates trigonometric concepts with algebra.

Rectangular Form

Rectangular form is another way to express complex numbers, typically presented as \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part. This representation visualizes the complex number as a point in the plane:

• Real part (a): Corresponds to the x-coordinate in the plane.
• Imaginary part (b): Corresponds to the y-coordinate in the plane.

To convert from polar to rectangular form, you leverage the trigonometric relationships:- \( a = r \cos \theta \)- \( b = r \sin \theta \)Given a polar form \( r(\cos \theta + i\sin \theta) \), you can easily convert it into \( a + bi \) form, making it straightforward and familiar, especially when dealing with addition or subtraction of complex numbers. This form is notably efficient for visualizing and calculating real-world applications involving both components independently.

Square Roots of Complex Numbers

Finding square roots of complex numbers is an intriguing task that reveals the beauty of complex analysis. The formula used is:\[z^{1/n} = r^{1/n}\left(\cos\left(\frac{\theta + 2k\pi}{n}\right) + i\sin\left(\frac{\theta + 2k\pi}{n}\right)\right)\]where \( n \) is the degree of the root, \( r \) the modulus, \( \theta \) the argument, and \( k \) any integer to find different roots.

When you are searching for square roots (i.e., \( n = 2 \)) of a complex number, such as \( z = 1 + i\sqrt{3} \), it typically has two roots. Here’s how they are determined:

• The modulus \( r^{1/2} \) gives the absolute size of the roots.
• Different values of \( k \) (like 0 and 1) calculate different angles, leading to distinct roots.

Calculating square roots in the polar form simplifies these operations significantly and can make seemingly complex problems more approachable. The roots are often then converted to rectangular form to ensure they are easily interpreted and applied in various contexts.