Problem 35 - Precalculus with Limits

Question

In Exercises 33-42, find the standard form of the complex number. Then represent the complex number graphically. \(\sqrt{48}[\cos(-30^{\circ}) + i\ \sin(-30^{\circ})]\)

Step-by-Step Solution

Answer

The standard form of given complex number is \(24\sqrt{3} - 12i\). When graphically represented: on the x-axis (real plane), plot \(24\sqrt{3}\) and for the y-axis (imaginary plane), go down 12 units.

Step 1: Identify magnitude and angle

First, identify the magnitude and angle from the given polar form of the complex number. Here, the magnitude \( r \) is \(\sqrt{48}\) and the angle \( \phi \) is -30 degrees.

Step 2: Calculate the cos and sin values

Determine the cos and sin value of the angle. In a unit circle, \(\cos(-30^{\circ}) = \sqrt{3}/2\) and \(\sin(-30^{\circ}) = -1/2\).

Step 3: Convert to Standard Form

Use the formula \( r(\cos\phi + i\sin\phi) \) to get to standard form \(a + ib\), \(\sqrt{48}(\frac{\sqrt{3}}{2} - \frac{1}{2}i) = \sqrt{3}\sqrt{48} - \frac{\sqrt{48}}{2}i \). After simplifying, the standard form is \(24\sqrt{3} - 12i\).

Step 4: Graph the Complex Number

Plot the complex number on the complex plane with the x-axis representing the real part and the y-axis representing the imaginary part. The real part is \(24\sqrt{3}\) and the imaginary part is -12, so find this point on the corresponding complex plane.

Key Concepts

Standard Form of Complex Number

The standard form of a complex number is expressed as \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit, satisfying \( i^2 = -1 \). This form is vital as it allows for straightforward operations like addition, subtraction, multiplication, and division, much like real numbers. In the context of the given exercise, the process started by identifying the magnitude \( r \) and angle \( \phi \) from the polar form. By calculating the cosine and sine of the given angle, we transformed the polar form into the standard form, \( 24\sqrt{3} - 12i \) after simplifying. This simplification holds the key to algebraic manipulations and further applications in complex analysis or electrical engineering, where standard form is widely used.

Polar Form of Complex Number

Complex numbers can also be represented in polar form, which is written as \( r(\text{cos} \phi + i \text{sin} \phi) \) or \( r \text{cis} \phi \), where \( r \) is the magnitude (also called modulus) and \( \phi \) is the angle (or argument) with the positive direction of the x-axis. The polar form is particularly advantageous when dealing with multiplication and powers of complex numbers. It turns complex multiplication into a simple product of magnitudes and a sum of angles. For the exercise given, the complex number began in its polar form with magnitude \( \sqrt{48} \) and angle of -30 degrees. The student converted this into the standard form, showcasing an essential skill in handling complex numbers.

Graphical Representation of Complex Numbers

Visualizing complex numbers on a graph involves plotting them on the complex plane, also known as the Argand diagram, which has a horizontal axis (real part) and a vertical axis (imaginary part). The location of any complex number can be determined by its standard form for easy spotting. When representing \( 24\sqrt{3} - 12i \) graphically, as shown in the exercise, one would mark a point where the real component \( 24\sqrt{3} \) meets the imaginary component \( -12 \) on their respective axes. This graphical perspective is crucial for understanding complex number operations, such as addition, where vectors are joined head-to-tail, or multiplication, where vectors are stretched and rotated.

De Moivre's Theorem

De Moivre's Theorem is a powerful formula in complex analysis, stating that for any real number \( \theta \) and positive integer \( n \), the expression \( (\cos \theta + i\sin \theta)^n \) is equivalent to \( \cos(n\theta) + i\sin(n\theta) \). This theorem simplifies the calculation of powers of complex numbers by converting them into a product of their polar form's magnitude to the power of \( n \) and the angle multiplied by \( n \). In practice, De Moivre's Theorem enables us to easily find the nth roots of a complex number, demonstrate trigonometric identities, and solve complex polynomials. Although not explicitly used in the given exercise, understanding this theorem enriches the student's ability to work with complex numbers in their polar forms and beyond.