Problem 8 - A Graphical Approach to Precalculus with Limits

Question

Find each power. Write the answer in rectangular form. Do not use a calculator. $$\left[3 \text { cis } 40^{\circ}\right]^{3}$$

Step-by-Step Solution

Answer

\(-13.5 + 13.5\sqrt{3}i\)

Step 1: Understanding the expression

The given expression is \( \left[3 \text{ cis } 40^{\circ}\right]^{3} \). Here, \( 3 \text{ cis } 40^{\circ} \) is in polar form where 3 is the modulus and \( 40^{\circ} \) is the argument or angle.

Step 2: Applying De Moivre's Theorem

De Moivre's Theorem states that for a complex number in polar form \( r \text{ cis } \theta \), the n-th power \( \left(r \text{ cis } \theta \right)^{n} = r^{n} \text{ cis } (n\theta) \). Here, \( r = 3 \), \( \theta = 40^{\circ} \), and \( n = 3 \).

Step 3: Calculating the new modulus

Raise the modulus to the power of 3: \( 3^{3} = 27 \).

Step 4: Calculating the new argument

Multiply the angle by 3: \( 3 \times 40^{\circ} = 120^{\circ} \).

Step 5: Expressing in rectangular form

Convert \( 27 \text{ cis } 120^{\circ} \) to rectangular form using \( a + bi = r(\cos \theta + i \sin \theta) \). Compute \( \cos 120^{\circ} = -0.5 \) and \( \sin 120^{\circ} = \frac{\sqrt{3}}{2} \). Thus, \( 27(\cos 120^{\circ} + i \sin 120^{\circ}) = 27(-0.5 + i \cdot \frac{\sqrt{3}}{2}) = -13.5 + 13.5\sqrt{3}i \).

Key Concepts

Complex Numbers

Complex numbers are a fascinating area in mathematics that extend the idea of numbers beyond the real number line. They are of the form \(a + bi\), where \(a\) is the real part, \(b\) is the imaginary part, and \(i\) is the imaginary unit defined as \(i^2 = -1\).

These numbers are incredibly useful in various fields such as engineering, physics, and computer science. They allow us to represent and manipulate data that has both a magnitude and direction.

• The real part \(a\) determines the horizontal position on the complex plane.
• The imaginary part \(b\) determines the vertical position on the complex plane.

Complex numbers make it easier to understand phenomena that have both "static" (real) and "dynamic" (imaginary) components.

Polar Form

Polar form offers a compact way to represent complex numbers, especially when considering operations like multiplication and finding powers. This form is expressed as \(r \text{ cis } \theta\) or \(r(\cos \theta + i \sin \theta)\), where \(r\) is the modulus and \(\theta\) represents the angle or argument.

The modulus \(r\) is calculated as \(\sqrt{a^2 + b^2}\), reflecting the distance from the origin or "magnitude."

• \(\theta\) is the angle formed with the positive real axis in the complex plane, often measured in radians or degrees.
• It represents the "direction" of the complex number from the origin.

Polar form is particularly beneficial when using De Moivre's Theorem, as it simplifies raising complex numbers to powers by only needing to manipulate \(r\) and \(\theta\).

Rectangular Form

Rectangular form, also known as Cartesian form, represents a complex number as \(a + bi\), where \(a\) and \(b\) are real numbers. This form is straightforward and quite intuitive when visualizing complex numbers on the complex plane.

• The real component \(a\) is its projection on the horizontal axis.
• The imaginary component \(b\) is its projection on the vertical axis.

While polar form excels with multiplication and exponentiation, rectangular form is preferred when adding or subtracting complex numbers. It allows easy calculation using basic arithmetic operations and provides a simple geometric interpretation.

Trigonometric Functions

Trigonometric functions \(\cos\) and \(\sin\) are crucial in converting complex numbers between polar and rectangular forms. They connect angles and side lengths of right triangles to complex numbers on the unit circle.

• \(\cos \theta\) gives the horizontal component, equivalent to the \(x\)-coordinate on the complex plane.
• \(\sin \theta\) provides the vertical component, equivalent to the \(y\)-coordinate.

Using these, we can express any complex number \(r\text{ cis }\theta\) in rectangular form as \(r(\cos \theta + i \sin \theta)\). This conversion is crucial for visualizing complex operations and finding rectangular representations. Trigonometric functions are also pivotal in computing with complex numbers, giving insight into their behavior over rotations and scaling.