Problem 71 - Precalculus Mathematics for Calculus

Question

Find the indicated power using De Moivre’s Theorem. $$(2 \sqrt{3}+2 i)^{5}$$

Step-by-Step Solution

Answer

The power is \(-512\sqrt{3} + 512i\).

Step 1: Convert to Polar Form

First, identify the real and imaginary parts of the complex number: The real part is \(2\sqrt{3}\) and the imaginary part is \(2i\). Calculate the modulus \(r\) as: \[ r = \sqrt{(2\sqrt{3})^2 + (2)^2} = \sqrt{12 + 4} = \sqrt{16} = 4 \] Next, find the argument \(\theta\): \[ \theta = \tan^{-1}\left(\frac{2}{2\sqrt{3}}\right) = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = 30^\circ \text{ or } \frac{\pi}{6} \text{ radians} \] Therefore, the complex number in polar form is \(4\left(\cos\frac{\pi}{6} + i\sin\frac{\pi}{6}\right)\).

Step 2: Apply De Moivre's Theorem

According to De Moivre's Theorem, \[ (r(\cos \theta + i\sin \theta))^n = r^n (\cos(n\theta) + i\sin(n\theta)) \] Where \(n\) is the power we want to raise the complex number to. For this exercise, \(n = 5\), \(r = 4\), and \(\theta = \frac{\pi}{6}\). Calculate \(r^n:\) \[ 4^5 = 1024 \] Then multiply the argument by 5: \[ 5 \times \frac{\pi}{6} = \frac{5\pi}{6} \] Thus, the expression becomes \[ 1024 \left(\cos\frac{5\pi}{6} + i\sin\frac{5\pi}{6}\right) \]

Step 3: Simplify Using Trigonometric Values

Calculate the trigonometric values: \[ \cos\frac{5\pi}{6} = -\cos\frac{\pi}{6} = -\frac{\sqrt{3}}{2} \] \[ \sin\frac{5\pi}{6} = \sin\frac{\pi}{6} = \frac{1}{2} \] Substitute these values into the expression: \[ 1024 \left(-\frac{\sqrt{3}}{2} + i\frac{1}{2}\right) = 1024 \left(-\frac{\sqrt{3}}{2}\right) + 1024 \left(i\frac{1}{2}\right) \] Simplify this to get: \[ -1024 \times \frac{\sqrt{3}}{2} + 1024i \times \frac{1}{2} = -512\sqrt{3} + 512i \]

Key Concepts

complex numbers

Complex numbers are a fascinating mathematical concept that allows us to extend the idea of numbers beyond the familiar realm of real numbers. They are composed of two parts: a real part and an imaginary part. A typical complex number is written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, defined by the property \(i^2 = -1\).
Some important points to remember about complex numbers:

• The real part \(a\) is simply a real number.
• The imaginary part \(bi\) involves the imaginary unit \(i\).
• Complex numbers can be represented on the complex plane, with the horizontal axis representing the real part and the vertical axis representing the imaginary part.
• Adding and subtracting complex numbers involves combining their real and imaginary parts separately.
• Multiplying complex numbers involves using the distributive property, making sure to apply the rule \(i^2 = -1\) when necessary.

Working with complex numbers opens up new possibilities in mathematics, allowing us to solve equations that have no solutions in the realm of real numbers alone.

polar form

The polar form of a complex number is another way to express it, emphasizing its magnitude and direction rather than its horizontal and vertical components. This is particularly useful in problems involving powers and roots of complex numbers.
The conversion to polar form involves two main pieces of information:

• The magnitude or modulus \(r\), which represents the distance of the complex number from the origin of the complex plane. It is calculated using the formula \(r = \sqrt{a^2 + b^2}\).
• The argument or angle \(\theta\), which tells us the direction of the complex number from the positive real axis. It is determined using \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\).

In polar form, a complex number \(a + bi\) is expressed as \(r(\cos \theta + i\sin \theta)\). This form is particularly helpful when raising complex numbers to a power, as seen in De Moivre's Theorem, making calculations more straightforward and elegant.

trigonometric functions

Trigonometric functions include sine, cosine, and tangent, among others. They have crucial roles not only in trigonometry but also in the polar form representation of complex numbers.
Here's how trigonometric functions apply to complex numbers:

• The sine function \(\sin\theta\) measures the vertical component of a complex number in polar form.
• The cosine function \(\cos\theta\) measures the horizontal component of a complex number in polar form.
• Combining these via \(\cos \theta + i\sin \theta\) allows us to describe any complex number in terms of its angle \(\theta\) and magnitude \(r\).

With De Moivre's Theorem, trigonometric functions enable us to compute powers of complex numbers simply by manipulating angles and magnitudes. For example, raising a complex number to a power causes the angle \(\theta\) to be multiplied by that power, known as \(n\), and the modulus \(r\) is raised to the power \(n\). This is a powerful way to handle complex number manipulation in many mathematical scenarios.