Problem 15 - College Trigonometry

Question

Find a polar representation for the complex number \(z\) and then identify \(\operatorname{Re}(z)\), \(\operatorname{Im}(z),|z|, \arg (z)\) and \(\operatorname{Arg}(z)\). $$ z=-7+24 i $$

Step-by-Step Solution

Answer

Real part: -7, Imaginary part: 24, Magnitude: 25, Argument \( \pi + \tan^{-1} \left( \frac{24}{7} \right) \).

Step 1: Identify the Real and Imaginary Parts

The complex number is given as \( z = -7 + 24i \). Identify the real part \( \operatorname{Re}(z) = -7 \) and the imaginary part \( \operatorname{Im}(z) = 24 \).

Step 2: Calculate the Magnitude

The magnitude \( |z| \) of a complex number \( z = a + bi \) is given by \( |z| = \sqrt{a^2 + b^2} \). Here, \( a = -7 \) and \( b = 24 \). So, \( |z| = \sqrt{(-7)^2 + (24)^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \).

Step 3: Determine the Argument

The argument \( \arg(z) \) of a complex number is given by \( \arg(z) = \tan^{-1} \left( \frac{b}{a} \right) \). Using \( a = -7 \) and \( b = 24 \), \( \arg(z) = \tan^{-1} \left( \frac{24}{-7} \right) \). Since \( z \) is in the second quadrant, the principal argument \( \operatorname{Arg}(z) = \pi + \tan^{-1} \left( \frac{24}{7} \right) \).

Step 4: Construct Polar Representation

The polar form of a complex number is \( z = |z| (\cos \theta + i \sin \theta) \). We have \( |z| = 25 \) and \( \theta = \pi + \tan^{-1} \left( \frac{24}{7} \right) \). So the polar representation is \( z = 25 \left( \cos \left(\pi + \tan^{-1} \left( \frac{24}{7} \right)\right) + i \sin \left(\pi + \tan^{-1} \left( \frac{24}{7} \right)\right) \right) \).

Key Concepts

Real and Imaginary Parts

Complex numbers have two main components - the real part and the imaginary part. When working with a complex number like \( z = -7 + 24i \), the real part, often denoted as \( \operatorname{Re}(z) \), is simply the coefficient of the real number, which in this case is \(-7\).
The imaginary part, denoted as \( \operatorname{Im}(z) \), is the coefficient of the imaginary unit \( i \), here being \(24\). This is essential because understanding the separation of real and imaginary components is the first step to analyzing complex numbers.

Magnitude of a Complex Number

The magnitude (or modulus) of a complex number gives a measure of its size. For a complex number \( z = a + bi \), the magnitude is given by the formula:

• \( |z| = \sqrt{a^2 + b^2} \)

Here, substituting \( a = -7 \) and \( b = 24 \), you get:

• \( |z| = \sqrt{(-7)^2 + (24)^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \)

This shows you the length of the line from the origin to the point \((-7, 24)\) in the complex plane. It is a crucial step because the magnitude is a key component of the polar form of a complex number.

Argument of a Complex Number

The argument of a complex number is the angle that the line representing the number makes with the positive real axis. Calculating this angle can be achieved through the formula:

• \( \arg(z) = \tan^{-1} \left( \frac{b}{a} \right) \)

In the case of \( z = -7 + 24i \), you calculate:

• \( \arg(z) = \tan^{-1} \left( \frac{24}{-7} \right) \)

Because our point is in the second quadrant, more adjustment is needed, where:

• \( \operatorname{Arg}(z) = \pi + \tan^{-1} \left( \frac{24}{7} \right) \)

Understanding the argument helps to identify the direction of the vector in the complex plane, which completes the information needed for polar representation.

Polar Form of a Complex Number

The polar form of a complex number is a way of expressing it using its magnitude and argument. For any complex number \( z = a + bi \), the polar form is:

• \( z = |z| (\cos \theta + i \sin \theta) \)

When you know the magnitude \( |z| \) and the argument \( \theta \), you can represent it as:

• Here \( |z| = 25 \) and \( \theta = \pi + \tan^{-1} \left( \frac{24}{7} \right) \)
• Thus, the polar representation is \( z = 25 \left( \cos \left( \pi + \tan^{-1} \left( \frac{24}{7} \right) \right) + i \sin \left( \pi + \tan^{-1} \left( \frac{24}{7} \right) \right) \right) \)

Using the polar form simplifies complex multiplication and division and provides an insightful geometric interpretation of complex numbers in terms of their magnitude and direction.