Problem 43 - Precalculus with Limits

Question

In Exercises 43-46, use a graphing utility to represent the complex number in standard form. \(5 \left(\cos\ \dfrac{\pi}{9} + i\ \sin\ \dfrac{\pi}{9} \right)\)

Step-by-Step Solution

Answer

The complex number in standard form is \(5 \cos(\dfrac{\pi}{9}) + 5i \sin(\dfrac{\pi}{9}) \), and the representation of the point on the graph with coordinates \((5 \cos(\dfrac{\pi}{9}), 5 \sin(\dfrac{\pi}{9}))\) corresponds to the given complex number.

Step 1: Calculate Real Part (a)

The real part \(a\) is given by \(a = r \cdot cos(\theta)\), where \(r\) is the modulus and \(\theta\) is the argument. For this exercise, \(r = 5\) and \(\theta = \dfrac{\pi}{9}\). Therefore, \(a = 5 \cos(\dfrac{\pi}{9})\).

Step 2: Calculate Imaginary Part (b)

The imaginary part, \(b\) is given by \(b = r \cdot sin(\theta)\). Therefore, \(b = 5 \sin(\dfrac{\pi}{9})\).

Step 3: Write the Complex Number in Standard Form

The standard form of a complex number is \(a + bi\). Using the computed values of \(a\) and \(b\) from Steps 1 and 2, the complex number in standard form will be \(5 \cos(\dfrac{\pi}{9}) + 5i \sin(\dfrac{\pi}{9}) \)

Step 4: Graph the Complex Number

Represent the complex number on the complex plane. Place the real part \(a\) on the x–axis and the imaginary part \(b\) on the y–axis. So graph a point on the complex plane corresponding to \((5 \cdot \cos(\dfrac{\pi}{9}), 5 \cdot \sin(\dfrac{\pi}{9}))\), that represents the complex number in the graphical form.

Key Concepts

Graphing Complex Numbers

Visualizing complex numbers on a plane offers a way to understand their properties and operations more intuitively. The complex plane, also known as the Argand diagram, has a horizontal axis (the real axis) and a vertical axis (the imaginary axis). Each complex number corresponds to a unique point on this plane. To graph a complex number, you simply locate the real part (a) on the x-axis and the imaginary part (b) on the y-axis.

For the given complex number in trigonometric form, you'd first convert it to standard form: the real part, obtained by multiplying the modulus by the cosine of the angle, and the imaginary part, found by the modulus times the sine of the angle. In the example \(5 \left(\cos\dfrac{\pi}{9} + i \sin\dfrac{\pi}{9} \right)\), this process entails plotting the point corresponding to \( (5 \cdot \cos(\dfrac{\pi}{9}), 5 \cdot \sin(\dfrac{\pi}{9})) \). This point represents our complex number graphically.

By graphing complex numbers, one can easily perform operations like addition and multiplication by visual means or observe properties such as conjugates, which are reflected across the real axis.

Standard Form of Complex Numbers

The standard form is the most common way to express a complex number. It is written as \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part, with \(i\) being the imaginary unit satisfying \(i^2 = -1\). To convert a complex number from trigonometric to standard form, we multiply the modulus (in our case, 5) by the cosine of the angle for the real part, and by the sine of the angle for the imaginary part.

From the given problem, we use this strategy to find \( a = 5 \cos(\dfrac{\pi}{9}) \) and \( b = 5 \sin(\dfrac{\pi}{9})\). Thus, the standard form of our complex number becomes \(5 \cos(\dfrac{\pi}{9}) + 5i \sin(\dfrac{\pi}{9})\). This form is instrumental in performing algebraic operations on complex numbers and is the baseline for understanding other forms and operations.

Trigonometric Form of Complex Numbers

The trigonometric form of a complex number expresses it using a modulus and an angle. This form is particularly useful for understanding complex numbers geometrically and for performing multiplications and finding powers or roots with relative ease. The general formula for converting a complex number to trigonometric form is \( r (\cos(\theta) + i \sin(\theta))\), where \(r\) is the modulus and \(\theta\) is the argument of the complex number.

In the exercise, the complex number is already in trigonometric form: \(5 \left(\cos\dfrac{\pi}{9} + i \sin\dfrac{\pi}{9} \right)\). Here, the modulus \(r\) is 5, and the angle \(\theta\) is \(\dfrac{\pi}{9}\). Knowing the trigonometric form allows us to easily revert to the standard form for basic operations or graphing, as well as to leverage De Moivre's theorem for complex number calculus involving powers and roots.