Problem 29 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ -20 i $$

Step-by-Step Solution

Answer

The trigonometric form of \(-20i\) is \(20(\cos(\frac{3\pi}{2}) + i\sin(\frac{3\pi}{2}))\).

Step 1: Identify the Real and Imaginary Parts

The given complex number is \(-20i\). This can be written in the form \(a + bi\), where \(a = 0\) and \(b = -20\).

Step 2: Calculate the Magnitude

Use the formula to find the magnitude (or modulus) of the complex number: \(r = \sqrt{a^2 + b^2}\).Substituting the values:\(r = \sqrt{0^2 + (-20)^2} = \sqrt{400} = 20\).

Step 3: Determine the Angle \(\theta\)

Since the imaginary part is negative and the real part is zero, the complex number lies on the negative imaginary axis. This gives \(\theta = \frac{3\pi}{2}\), as it points directly downward on the complex plane from the origin.

Step 4: Express in Trigonometric Form

The trigonometric form of a complex number is \(r(\cos(\theta) + i\sin(\theta))\).Substitute the magnitude and angle:\(20(\cos(\frac{3\pi}{2}) + i\sin(\frac{3\pi}{2}))\).

Key Concepts

Trigonometric Form

To understand the **trigonometric form** of complex numbers, imagine the complex numbers as points in a 2D plane. This plane has a horizontal axis called the real axis and a vertical axis called the imaginary axis. The trigonometric form helps to express a complex number in terms of its distance from the origin and the angle it makes with the positive real axis.

For any complex number in the form of \(a + bi\), its trigonometric form is given by:

• Magnitude \(r = \sqrt{a^2 + b^2}\)
• Angle \(\theta\) is the angle with the positive real axis (can be found using trigonometric identities such as arctan and considerations of the quadrants).

Thus, a complex number can be written as \(r(\cos(\theta) + i\sin(\theta))\). This representation is particularly useful for simplifying multiplication and division of complex numbers and provides a geometric interpretation.

Imaginary Axis

The **imaginary axis** is the vertical line in the complex plane. It is key in understanding the position of complex numbers. When a complex number is placed on the imaginary axis, it means that its real part is zero. Such numbers are purely imaginary, and their placement on the imaginary axis depends entirely on the imaginary component.

For example, in the complex number \(-20i\):

• The real part is \(0\)
• The imaginary part is \(-20\)

This complex number lies entirely on the negative side of the imaginary axis. Understanding this layout helps in determining the angle \(\theta\) for trigonometric notation, especially when the number points downwards or upwards from the origin.

Magnitude of Complex Numbers

The **magnitude of complex numbers**, also known as the modulus, is essentially the distance from the origin to the point representing the complex number in the complex plane. It is analogous to the length of the vector from the origin to this point.

• For a complex number expressed as \(a + bi\), its magnitude \(r\) is calculated using the formula \(r = \sqrt{a^2 + b^2}\).

The magnitude is a crucial part of expressing a complex number in trigonometric form because it scales the unit circle representation of the number.

In our example, the magnitude of \(-20i\) is \(20\), calculated as \(\sqrt{0^2 + (-20)^2}\). The magnitude is positive regardless of the signs of its components, emphasizing the distance and not the direction on the imaginary axis.