Problem 5 - A Graphical Approach to Precalculus with Limits

Question

Find each power. Write the answer in rectangular form. Do not use a calculator. $$\left[2\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right)\right]^{3}$$

Step-by-Step Solution

Answer

The rectangular form is 8.

Step 1: Identify Basic Form

The given problem is finding the power of a complex number in polar form, where the complex number is presented as \(z = r(\cos\theta + i\sin\theta)\). Here, \(r = 2\) and \(\theta = \frac{2\pi}{3}\).

Step 2: Apply De Moivre's Theorem

De Moivre's Theorem tells us that to compute \(z^n\), we use: \[(r(\cos\theta + i\sin\theta))^n = r^n(\cos(n\theta) + i\sin(n\theta))\]. Substitute \(r = 2\), \(n = 3\), and \(\theta = \frac{2\pi}{3}\), we get: \[z^3 = (2^3)(\cos(3\cdot \frac{2\pi}{3}) + i\sin(3\cdot \frac{2\pi}{3}))\].

Step 3: Calculate the Power of the Magnitude

Compute \(r^n\), which is \(2^3 = 8\). So, \(z^3 = 8(\cos(3\cdot \frac{2\pi}{3}) + i\sin(3\cdot \frac{2\pi}{3}))\).

Step 4: Calculate the Multiple of the Angle

Calculate \(3\cdot \frac{2\pi}{3} = 2\pi\). Since \(\cos(2\pi) = 1\) and \(\sin(2\pi) = 0\), we have: \(\cos(3\cdot \frac{2\pi}{3}) = 1\) and \(\sin(3\cdot \frac{2\pi}{3}) = 0\).

Step 5: Simplify to Get Rectangular Form

Substituting the values from Step 4 into the expression from Step 3, we get: \[8(\cos(3\cdot \frac{2\pi}{3}) + i\sin(3\cdot \frac{2\pi}{3})) = 8(1 + i\cdot 0) = 8 + 0i = 8.\] Thus, the rectangular form is \(8\).

Key Concepts

polar form

Polar form is a way to represent complex numbers, which provides an intuitive understanding about their magnitude and direction. In this form, the complex number is expressed as \( r(\cos \theta + i\sin \theta) \), where:

• \( r \) is the magnitude (or modulus) of the number.
• \( \theta \) is the angle (or argument) from the positive x-axis to the line representing the complex number.

Polar form is particularly useful for multiplication and division of complex numbers and is the basis for De Moivre's Theorem, which simplifies computation of powers and roots. For example, the complex number given in the exercise \( 2(\cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3}) \) uses this form, where \( r = 2 \) and \( \theta = \frac{2\pi}{3} \).
Understanding polar form is key to applying trigonometric identities and transformations to complex numbers. It allows you to work with angles and magnitudes directly, making it simpler to solve complex multiplication and exponentiation tasks.

complex numbers

Complex numbers are numbers that include both a real part and an imaginary part. They are often presented in the form \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part, with \( i \) representing the square root of \(-1\).
These numbers are visualized as points in a two-dimensional plane known as the complex plane, with the real part along the x-axis and the imaginary part along the y-axis.

• The real part (\( a \)) represents the horizontal displacement.
• The imaginary part (\( b \)) represents the vertical displacement.

Complex numbers are vital in various fields such as engineering, physics, and applied mathematics. They allow for the analysis of sinusoidal waveforms and other oscillatory motion. By converting complex numbers to polar form, operations like exponentiation become more straightforward, as seen with De Moivre’s Theorem.
In the given problem, the complex number was manipulated using polar coordinates to simplify the calculation of powers.

rectangular form

The rectangular form, also known as the standard or Cartesian form, is another way to express complex numbers. It writes a complex number as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part.
In this representation, complex arithmetic becomes similar to vector operations in two dimensions. Addition and subtraction are straightforward:

• Add the real parts together.
• Add the imaginary parts together.

However, multiplication and division can be less intuitive, which is why converting to polar form can be beneficial for those operations.
For this exercise, the problem requested the result in rectangular form. After applying De Moivre’s Theorem to find the power of the complex number, the solution was converted back to rectangular form. This resulted in \( 8 + 0i \), simplifying to \( 8 \), which purely represents a real number on the real axis of the complex plane.