Problem 38 - Precalculus with Limits

Question

In Exercises 33-42, find the standard form of the complex number. Then represent the complex number graphically. \(6 \left(\cos\ \dfrac{5\pi}{12} + i\ \sin\ \dfrac{5\pi}{12} \right)\)

Step-by-Step Solution

Answer

The standard form of the given complex number is \( a+bi \), where \( a= 6\cos\ \dfrac{5\pi}{12} \) and \( b= 6\sin\ \dfrac{5\pi}{12} \). It can be represented graphically by plotting the point \( (a, b) \) on the complex plane.

Step 1: Calculation of Real Part

Calculate the value of \( \cos\ \dfrac{5\pi}{12} \) and then multiply it by the modulus, here 6, to find the real part \( a \) of the complex number. So, \( a= 6\cos\ \dfrac{5\pi}{12} \)

Step 2: Calculation of Imaginary Part

Calculate the value of \( \sin\ \dfrac{5\pi}{12} \) and then multiply it by the modulus, here 6, to find the imaginary part \( b \) of the complex number. So, \( b= 6\sin\ \dfrac{5\pi}{12} \)

Step 3: Write in Standard Form

Write the complex number in the standard form \( a+b i \)

Step 4: Graphical Representation

To represent this complex number graphically, plot the point \( (a, b) \) on the complex plane, where the x-axis is the real axis and the y-axis is the imaginary axis

Key Concepts

Standard Form of Complex Number

The standard form of a complex number is written as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, with \( i \) representing the square root of -1. This is the most widely used form for complex numbers as it clearly separates the real and imaginary components, making calculations and understanding easier. In the exercise provided, we convert a complex number from its trigonometric form into this standard form. By calculating the cosine and sine of given angles and then multiplying these by the modulus, we obtain the real and imaginary parts, respectively.

Graphical Representation of Complex Number

Plotting a complex number on a plane gives us a visual understanding of its properties. The horizontal axis, typically known as the real axis, represents the real part of a complex number, while the vertical axis, or the imaginary axis, represents the imaginary part. This plane is known as the complex plane or Argand plane. To graphically represent the complex number given in the trigonometric form in the exercise, you calculate the real part (\( a \)) and the imaginary part (\( b \)) and then plot the point \( (a, b) \). This representation helps in visualizing complex number operations, such as addition, as geometrical translations.

Trigonometric Form of Complex Number

The trigonometric form of a complex number is given by \( r(\text{cos} \theta + i\text{sin} \theta) \), where \( r \) is the modulus of the complex number, and \( \theta \) is the argument or angle formed with the positive direction of the real axis. This form is especially useful for multiplying, dividing, and raising complex numbers to powers. In the exercise problem, we start with a complex number already in trigonometric form, and our goal is to simplify it into its standard form. Understanding this conversion is key to working effectively with complex numbers across different representations.

Calculating Real and Imaginary Parts

Finding the real and imaginary parts of a complex number is fundamental to nearly all operations involving complex numbers. The real part, \( a \), is obtained by multiplying the modulus by the cosine of the angle, while the imaginary part, \( b \), is obtained by multiplying the modulus by the sine of the angle. In our exercise, the real and imaginary parts are calculated from a trigonometric expression. This involves evaluating the cosine and sine of the specified angle, \( \frac{5\tau}{12} \), and then scaling these values by the modulus, which effectively tells us how far the number is from the origin in the complex plane.