Problem 80 - Precalculus with Limits

Question

Finding a Power of a Complex Number In Exercises \(65-80\) , use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$\left[2\left(\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}\right)\right]^{6}$$

Step-by-Step Solution

Answer

The 6th power of \(2(\cos \frac{\pi}{8}+i \sin \frac{\pi}{8})\) in standard form is \( -64\sqrt{2}+64\sqrt{2}i \)

Step 1: Identify r, \(\theta\) and n

From the problem, extract the values of r, \(\theta\) and n. In this case, \(r=2\), \(\theta=\frac{\pi}{8}\) and \(n=6\)

Step 2: Apply DeMoivre's Theorem

Substitute the values of \(r\), \(\theta\), and \(n\) into DeMoivre's theorem. This gives: \[ (2 (\cos\frac{\pi}{8} + isin\frac{\pi}{8}))^{6} = 2^{6} [cos(6*\frac{\pi}{8}) + i sin(6*\frac{\pi}{8})] \]

Step 3: Calculate the powers and multiples.

Calculate \(2^6=64\) and \(6*\frac{\pi}{8}= \frac{3\pi}{4}\). Replace these in the equation: \[ 2^{6} [cos(6*\frac{\pi}{8}) + i sin(6*\frac{\pi}{8})] = 64[cos(\frac{3\pi}{4}) + i sin(\frac{3\pi}{4})] \]

Step 4: Transform to the standard form

Transform the trigonometric form into the standard rectangular form. We know that \(cos(\frac{3\pi}{4}) = - \frac{1}{\sqrt{2}}\) and \(sin(\frac{3\pi}{4}) = \frac{1}{\sqrt{2}}\). Substitute these into the expression to get: \[64[-\frac{1}{\sqrt{2}} + i*\frac{1}{\sqrt{2}}]\] Simplify this to obtain \( -64\sqrt{2} + i*64\sqrt{2}\)

Key Concepts

Complex Numbers

Complex numbers are numbers that have two parts: a real part and an imaginary part. The standard form of a complex number is expressed as \( a + bi \), where \( a \) is the real part, and \( bi \) is the imaginary part.
Here, \( i \) is the imaginary unit, which is equal to the square root of -1. Therefore, \( i^2 = -1 \).
Complex numbers extend the concept of the one-dimensional number line to the complex plane by using the real part as the x-axis and the imaginary part as the y-axis.

• Real numbers are complex numbers with 0 as the imaginary part (e.g., 5 is 5 + 0i).
• Imaginary numbers are complex numbers with 0 as the real part (e.g., 3i is 0 + 3i).

Understanding complex numbers is essential for working with electrical engineering, quantum mechanics, and many areas of mathematics.

Polar Form

The polar form is a way to represent complex numbers, which makes operations like multiplication and finding powers easier. In this form, a complex number is expressed as \( r(\cos \theta + i \sin \theta) \), where \( r \) is the modulus and \( \theta \) is the argument.

• The modulus \( r \) is the distance from the origin to the point in the complex plane and is calculated as \( r = \sqrt{a^2 + b^2} \).
• The argument \( \theta \) is the angle measured from the positive x-axis to the line segment connecting the origin to the point.

Polar form is particularly useful for multiplying complex numbers and when using DeMoivre's Theorem, as it easily incorporates angles through trigonometric functions.

Standard Form

After finding a complex number's power or root in polar form, you often convert it back to standard form for clarity. The standard form refers to expressing a complex number as \( a + bi \).
After simplification, our complex number becomes \( -64\sqrt{2} + i*64\sqrt{2} \). This means:

• The real part is \( -64\sqrt{2} \)
• The imaginary part is \( 64\sqrt{2} \)

Converting between forms involves understanding both algebraic manipulation and geometrical representation, bridging the gap between an analytical approach and visualization.

Trigonometric Functions

Trigonometric functions like cosine and sine are crucial when converting complex numbers from polar to standard form. These functions help describe the angle and direction of the complex number in the complex plane.

• Cosine (\(\cos \theta\)) represents the horizontal coordinate of the point.
• Sine (\(\sin \theta\)) represents the vertical coordinate of the point.

In DeMoivre’s Theorem, used for powers and roots of complex numbers, we rely on these trigonometric functions to manage angles. For example, converting \( \cos(\frac{3\pi}{4}) \) and \( \sin(\frac{3\pi}{4}) \) to numerical values allows the transformation from polar to standard form. Working with these trigonometric identities can simplify calculations and provide clear insights into complex number operations.