Problem 38 - Precalculus with Limits

Question

Writing a Complex Number in Standard Form In Exercises \(31-40\) , write the standard form of the complex number. Then represent the complex number graphically. $$8\left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right)$$

Step-by-Step Solution

Answer

The standard form of the complex number is \(8i\). Its graphical representation is a point on the y-axis, 8 units above the origin.

Step 1: Convert to Standard Form

To convert the complex number from trigonometric form to standard form, which is \(a+bi\), we first change the cosine and sine to its values. The cosine of \(\frac{\pi}{2}\) is 0 and the sine of \(\frac{\pi}{2}\) is 1. So, the complex number becomes \(8(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}) = 8(0 + i(1)) = 8i\).

Step 2: Graphical Representation

To represent the complex number graphically, we consider the imaginary part (8) as the y-coordinate and the real part (which is 0 in this case) as the x-coordinate. So, the complex number \(8i\) is represented on the y-axis, 8 units above the origin.

Key Concepts

Standard Form of Complex Numbers

A complex number in standard form is expressed as \(a + bi\), where \(a\) and \(b\) are real numbers. Here, \(a\) represents the real part, while \(b\) represents the imaginary part, and \(i\) is the imaginary unit with the property that \(i^2 = -1\). In the example provided, the goal is to convert a trigonometric representation of a complex number into this standard form.

• The given complex number is \(8(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2})\).
• The cosine of \(\frac{\pi}{2}\) is 0 and the sine of \(\frac{\pi}{2}\) is 1.
• Substituting these values into the expression gives: \(8(0 + i(1)) = 8i\).
• Thus, the standard form of the complex number is \(0 + 8i\).

This process involves understanding trigonometric values and applying them to convert the number into a format that can easily be manipulated algebraically.

Graphical Representation of Complex Numbers

Graphically, complex numbers can be represented on a plane known as the complex plane or Argand plane. This plane is similar to a Cartesian plane, but it is specifically used for complex numbers. The horizontal axis (x-axis) is used for the real part, and the vertical axis (y-axis) is used for the imaginary part.

• For a complex number \(a + bi\), the point \((a, b)\) is plotted on this plane.
• In the example \(8i\), the real part is 0 and the imaginary part is 8.
• This means the point \((0, 8)\) is located 8 units above the origin on the y-axis.

Graphing complex numbers helps in visualizing their magnitude and direction, making it easier to interpret operations such as addition, subtraction, and finding conjugates.

Trigonometric Form of Complex Numbers

The trigonometric form of a complex number (also known as polar form) expresses complex numbers using a radius (magnitude) and an angle (argument), written as \(r(\cos \theta + i \sin \theta)\). It's particularly useful for multiplying and dividing complex numbers. In this form:

• \(r\) is the modulus or absolute value, calculated as \(\sqrt{a^2 + b^2}\) for a complex number \(a + bi\).
• \(\theta\) is the argument or angle, typically determined using the arctangent function: \(\theta = \tan^{-1}(\frac{b}{a})\).
• In the given complex number \(8(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2})\), \(r = 8\) and \(\theta = \frac{\pi}{2}\).

This form is an alternative to the standard form and underscores the geometric interpretation of complex numbers, defining both their size and their direction on the complex plane.