Problem 13 - 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 \(\operatorname{Arg}(z)\). $$ z=3+4 i $$

Step-by-Step Solution

Answer

The polar form is \( 5 (\cos(0.927) + i \sin(0.927)) \), \( \operatorname{Re}(z) = 3 \), \( \operatorname{Im}(z) = 4 \), \( |z| = 5 \), \( \arg(z) = 0.927 \), \( \text{Arg}(z) = 0.927 + 2k\pi \).

Step 1: Convert to Polar Form

A complex number in the form \( z = a + bi \) can be expressed in polar form as \( z = r ( ext{cos} heta + i ext{sin} heta) \), where \( r = \sqrt{a^2 + b^2} \) is the magnitude and \( \theta = \text{arctan}(b/a) \) is the argument. Here, \( a = 3 \) and \( b = 4 \). First calculate \( r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \).

Step 2: Calculate Argument (Principal Value)

The argument \( \theta \) can be found using \( \theta = \text{arctan}(b/a) = \text{arctan}(4/3) \). This gives \( \theta \approx 0.927 \) in radians. The polar form of \( z \) is thus \( 5 (\cos(0.927) + i \sin(0.927)) \).

Step 3: Identify the Real Part

The real part of the complex number \( z = 3 + 4i \) is simply the coefficient of the real portion. Therefore, \( \operatorname{Re}(z) = 3 \).

Step 4: Identify the Imaginary Part

The imaginary part of \( z = 3 + 4i \) is the coefficient of the imaginary portion, without the \( i \). Therefore, \( \operatorname{Im}(z) = 4 \).

Step 5: Identify Magnitude

The magnitude (or modulus) \( |z| \) of the complex number is the distance from the origin to the point \( (3, 4) \) in the complex plane. We previously computed \( |z| = \sqrt{3^2 + 4^2} = 5 \).

Step 6: Calculate Argument (All Values)

For the argument, you have the principal value \( \theta = 0.927 \) radians, however, any integer \( k \) can be added to a full circle rotation (\( 2\pi k \)) to get the principal argument, \( \text{Arg}(z) = 0.927 + 2k\pi \).

Key Concepts

Magnitude of a Complex Number

The magnitude of a complex number, often denoted as \(|z|\), is a measure of its size or length in the complex plane. For a complex number in the form \(z = a + bi\), where \(a\) is the real part and \(b\) is the imaginary part, the magnitude is calculated using the formula:
\[ |z| = \sqrt{a^2 + b^2} \]. This formula comes from applying the Pythagorean theorem to the right triangle formed by \(a\) and \(b\) on the complex plane, with \(|z|\) being the hypotenuse of that triangle.
Interestingly, the magnitude represents the distance between the origin \((0,0)\) and the point \((a,b)\) in the complex plane.
For the given example, \(z = 3 + 4i\), its magnitude, \(|z|\), is \(5\), calculated as:

• Step 1: Square the real part: \(3^2 = 9\)
• Step 2: Square the imaginary part: \(4^2 = 16\)
• Step 3: Sum the squares: \(9+16 = 25\)
• Step 4: Take the square root: \(\sqrt{25} = 5\)

Real and Imaginary Parts

Every complex number contains a real part and an imaginary part, which together define its position in the complex plane. For the complex number \(z = a + bi\):

• The real part is \(a\), represented as \(\operatorname{Re}(z)\).
• The imaginary part is \(b\), represented as \(\operatorname{Im}(z)\), but it is usually denoted without the \(i\).

In the complex number \(z = 3 + 4i\), \(\operatorname{Re}(z) = 3\) and \(\operatorname{Im}(z) = 4\).
These components are crucial because they allow us to express complex numbers in standard form and connect algebraically with phenomena in real analysis, engineering, and physics.
The real part varies along the x-axis (horizontal), and the imaginary part varies along the y-axis (vertical) on the complex plane. This orthogonal relationship provides a straightforward way to visually analyze and understand complex numbers.

Argument of a Complex Number

The argument of a complex number, denoted as \(\arg(z)\), represents the angle formed between the positive real axis and the line connecting the origin to the point \((a, b)\) in the complex plane. This angle is generally given in radians.
The argument can be determined from the formula:
\[ \theta = \arctan \left( \frac{b}{a} \right) \].
For the complex number \(z = 3 + 4i\), \(b = 4\) and \(a = 3\), leading to:

• Calculate \(\theta = \arctan(4/3)\)
• \(\theta \approx 0.927\) radians

It's important to note that this is just one interpretation of the argument. To find additional possible arguments (the principal values), we add integer multiples of a full circle's angle, \(2\pi\). This is expressed as:
\(\text{Arg}(z) = 0.927 + 2k\pi\), where \(k\) is any integer.
This property is what gives the complex plane its cyclic nature, akin to periodic phenomena in trigonometry.

Complex Numbers in Polar Form

Complex numbers can be elegantly expressed in polar form, providing a powerful alternative to the standard rectangular form \(z = a + bi\). The polar form utilizes the magnitude \(|z|\) and the argument \(\theta\) as:
\[ z = r(\cos\theta + i\sin\theta) \].
This representation serves as a vital tool, especially in simplifying the multiplication and division of complex numbers, akin to how polar coordinates simplify operations involving points in a plane.
For the complex number \(z = 3 + 4i\):

• Magnitude \(r = 5\), as previously calculated.
• Argument \(\theta \approx 0.927\)
• Polar form becomes \(z = 5(\cos(0.927) + i\sin(0.927))\)

This form highlights the rotational aspect of complex numbers, making it easier to handle complex transformations and rotations. It's widely used in fields such as physics and electrical engineering, where such operations are critical.