Problem 59 - College Trigonometry

Question

Use DeMoivre's Theorem to find the indicated power of the given complex number. Express your final answers in rectangular form. $$ \left(\frac{3}{2}-\frac{3}{2} i\right)^{3} $$

Step-by-Step Solution

Answer

The final answer is \( -\frac{27}{8} - \frac{27}{8} i \).

Step 1: Convert to Polar Form

Given complex number is \( z = \frac{3}{2} \cdot (1 - i) \). Convert this to polar form. First, calculate the modulus.\[ r = \sqrt{\left(\frac{3}{2}\right)^2 + \left(-\frac{3}{2}\right)^2} = \sqrt{\frac{9}{4} + \frac{9}{4}} = \sqrt{\frac{18}{4}} = \sqrt{\frac{9}{2}} = \frac{3\sqrt{2}}{2} \]Next, find the argument \( \theta \) using \( \tan \theta = \frac{-\frac{3}{2}}{\frac{3}{2}} = -1 \), which gives \( \theta = -\frac{\pi}{4} \). Therefore, the polar form is \( z = \frac{3\sqrt{2}}{2} \left( \cos \left(-\frac{\pi}{4}\right) + i \sin \left(-\frac{\pi}{4}\right) \right) \).

Step 2: Apply DeMoivre's Theorem

DeMoivre's Theorem states that for a complex number in polar form \( r(\cos \theta + i \sin \theta) \), the \( n \)-th power is \( r^n (\cos(n\theta) + i \sin(n\theta)) \). Substitute \( r = \frac{3\sqrt{2}}{2} \), \( \theta = -\frac{\pi}{4} \), and \( n = 3 \): \[\left(\frac{3\sqrt{2}}{2}\right)^3 \left( \cos \left(3 \times -\frac{\pi}{4}\right) + i \sin \left(3 \times -\frac{\pi}{4}\right) \right) = \frac{27\sqrt{2}}{8} \left( \cos \left(-\frac{3\pi}{4}\right) + i \sin \left(-\frac{3\pi}{4}\right) \right)\]

Step 3: Simplify Cosine and Sine Components

Evaluate the trigonometric expressions: \( \cos \left(-\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \) and \( \sin \left(-\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \). Substituting these values gives: \[ \frac{27\sqrt{2}}{8} \left( -\frac{\sqrt{2}}{2} + i \left(-\frac{\sqrt{2}}{2}\right) \right) = \frac{27\sqrt{2}}{8} \times \left( -\frac{\sqrt{2}}{2} \right) + i \frac{27\sqrt{2}}{8} \times \left(-\frac{\sqrt{2}}{2}\right) \]

Step 4: Convert Back to Rectangular Form

Multiply and simplify \( \frac{27\sqrt{2}}{8} \times -\frac{\sqrt{2}}{2} = \frac{27}{8}(i - 1) \). Distribute \( \frac{27}{8} \) over the binomial: \[ x = \frac{27}{8}(-1) = -\frac{27}{8}, \quad y = \frac{27}{8}(-1) = -\frac{27}{8} \]Thus, the result in rectangular form is \( -\frac{27}{8} - \frac{27}{8} i \).

Key Concepts

Polar Form

Polar form is a way to represent complex numbers, emphasizing their amplitude and angle relative to the real axis. It is rendered as: \( z = r(\cos \theta + i \sin \theta) \), where \( r \) is the modulus (size) of the complex number and \( \theta \) is the argument (angle).
To convert from rectangular to polar:

• Compute the modulus using the formula: \( r = \sqrt{x^2 + y^2} \), where \( x \) and \( y \) are the real and imaginary parts.
• Find the argument using \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \).

This form is especially useful when performing multiplications or powers of complex numbers, as operations can be greatly simplified using angles instead of components.

Rectangular Form

In rectangular form, a complex number is expressed with its components along the real and imaginary axes: \( z = x + yi \), where \( x \) represents the real part and \( y \) represents the imaginary part. This form aligns well with Cartesian coordinates.

Advantages of rectangular form include:

• Simplicity in adding and subtracting complex numbers by directly combining like terms.
• Easier visualization as plotted points in the complex plane, making magnitude or argument calculations straightforward with trigonometry.

While rectangular is intuitive for basic operations, polar form offers efficiency for others, such as multiplication or exponentiation, as demonstrated in complex power calculations.

Complex Numbers

Complex numbers extend the regular number system to include solutions to equations like \( x^2 + 1 = 0 \). They combine a real component and an imaginary component (\( i \)), written as \( a + bi \).

Key Points about complex numbers:

• The imaginary unit \( i \) is defined by \( i^2 = -1 \).
• They can be visualized on a plane with a horizontal real axis and vertical imaginary axis, facilitating understanding of their operations and transformations.
• Complex numbers are crucial in many fields such as engineering, physics, and applied mathematics.

Understanding complex numbers, and mastering their transformations between polar and rectangular forms, is pivotal for problems like calculating powers using DeMoivre's Theorem which simplify intricate calculations by leveraging properties of angles.