Problem 39 - Algebra and Trigonometry

Question

\(25-48=\) Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi .\) $$ 3+4 i $$

Step-by-Step Solution

Answer

Polar form is \(5 (\cos(0.93) + i\sin(0.93))\).

Step 1: Identify the Real and Imaginary Components

The complex number given is \(3 + 4i\). Here, the real part \(a = 3\) and the imaginary part \(b = 4\).

Step 2: Calculate the Magnitude

Use the formula for the magnitude \(r\) of a complex number, which is given by \(r = \sqrt{a^2 + b^2}\). Substitute the values: \(r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\).

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

The argument \(\theta\) is given by \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\). Substituting the values gives \(\theta = \tan^{-1}\left(\frac{4}{3}\right)\). Calculate \(\theta\) which is approximately 0.93 radians.

Step 4: Write the Polar Form

The polar form of a complex number is expressed as \(r (\cos(\theta) + i\sin(\theta))\). Substituting in the magnitude and argument calculated: \(5 \left( \cos(0.93) + i\sin(0.93) \right)\).

Key Concepts

Complex Number

A complex number is a type of number that includes a real part and an imaginary part. It is usually written in the form of \( a + bi \), where \( a \) is the real component and \( bi \) is the imaginary component. Here, \( i \) represents the imaginary unit, defined by \( i = \sqrt{-1} \). Understanding the components of a complex number is essential for performing operations such as addition, subtraction, multiplication, and division. Additionally, complex numbers are valuable in various fields like engineering, physics, and mathematics, especially when dealing with wave functions or oscillations. In the problem given, the complex number is \( 3 + 4i \). Here, \( a = 3 \) is the real part and \( b = 4 \) is the imaginary part.

Magnitude of a Complex Number

The magnitude (or modulus) of a complex number is a measure of its "size" or distance from the origin when plotted on the complex plane. To find the magnitude of a complex number \( a + bi \), you use the formula:

• Magnitude, \( r = \sqrt{a^2 + b^2} \)

This formula is derived from the Pythagorean theorem and calculates the Euclidean distance from the point \((a, b)\) to the origin \((0, 0)\). In our exercise, the complex number is \( 3 + 4i \). Calculate the magnitude by substituting \( a = 3 \) and \( b = 4 \) into the formula:

• \( r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5 \)

Thus, the magnitude of the complex number is 5.

Argument of a Complex Number

The argument of a complex number is the angle formed with the positive real axis on the complex plane. It provides directional information and is essential for expressing the complex number in polar form. The argument, usually denoted as \( \theta \), can be found using the arctangent function:

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

Here, \( b \) is the imaginary component and \( a \) is the real component of the complex number. The angle \( \theta \) is measured in radians and should fall between \( 0 \) and \( 2\pi \).For the complex number \( 3 + 4i \), the argument \( \theta \) can be found by:

• \( \theta = \tan^{-1}\left(\frac{4}{3}\right) \)

This calculation results in an angle of approximately 0.93 radians. This angle helps transform the complex number into polar form, offering a different perspective that can be useful in certain calculations and applications.