Problem 33 - Calculus

Question

The complex number \(1 /\left(r e^{i \theta}\right)\) has polar form _______ and rectangular form _______ .

Step-by-Step Solution

Answer

Polar form: \(\frac{1}{r} e^{-i \theta}\); Rectangular form: \(\frac{1}{r} \cos(\theta) - i \frac{1}{r} \sin(\theta)\).

Step 1: Understand the Given Complex Number in Polar Form

The given complex number is \(\frac{1}{r e^{i \theta}}\). A complex number in polar form is expressed as \( r e^{i \theta} \). Here, \(r\) is the modulus and \(\theta\) is the argument, or angle, of the complex number.

Step 2: Reciprocate the Complex Number

To find the reciprocal of a complex number in polar form \( r e^{i \theta} \), take the reciprocal of its modulus and negate its angle. Therefore, the reciprocal in polar form is \( \frac{1}{r} e^{-i \theta} \).

Step 3: Write in Polar Form

The polar form of \( \frac{1}{r e^{i \theta}} \) is \( \frac{1}{r} e^{-i \theta} \). This means the modulus is \( \frac{1}{r} \) and the argument is \(-\theta\).

Step 4: Convert Polar Form to Rectangular Form

To convert from polar to rectangular form, use the formula: \( z = r \cos(\theta) + i r \sin(\theta) \). For \( \frac{1}{r} e^{-i \theta} \), the rectangular form is \(\frac{1}{r} \cos(-\theta) + i \frac{1}{r} \sin(-\theta) \).

Step 5: Use Trigonometric Identities

Remember: \( \cos(-\theta) = \cos(\theta) \) and \( \sin(-\theta) = -\sin(\theta) \). Thus, the rectangular form becomes \( \frac{1}{r} \cos(\theta) - i \frac{1}{r} \sin(\theta) \).

Key Concepts

Polar Form

Complex numbers can be represented in various forms, with polar form being quite useful for various calculations. In polar form, a complex number is expressed as \( r e^{i \theta} \). Here, \( r \) is the modulus of the complex number, representing its magnitude or size. The angle \( \theta \) is the argument, which is the direction of the complex number on the complex plane.

This form is particularly beneficial when performing multiplication or finding powers of complex numbers as it simplifies the process. For example, multiplying two complex numbers in polar form simply involves multiplying their moduli and adding their arguments:

• Product of moduli: \( r_1 r_2 \).
• Sum of arguments: \( \theta_1 + \theta_2 \).

Understanding the relationship between polar form and other mathematical operations can provide a clearer grasp of complex numbers and their behavior.

Rectangular Form

Rectangular form, also known as Cartesian form, expresses a complex number as \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit. In this representation:

• \( a \) is the real part of the complex number.
• \( b \) is the imaginary part.

The main advantage of rectangular form is its straightforward application in addition and subtraction, as these operations become simple arithmetic functions:

• To add two complex numbers, add their real parts and imaginary parts separately.
• For subtraction, subtract the respective parts.

Converting between rectangular and polar forms is a common practice, especially when dealing with trigonometric functions and angles, making it a valuable skill in various fields of science and engineering.

Trigonometric Identities

Trigonometric identities are essential tools when working with complex numbers, especially when converting between polar and rectangular forms. They allow us to simplify expressions and perform calculations more easily. In the given exercise, we used two important trigonometric identities:

• \( \cos(-\theta) = \cos(\theta) \)
• \( \sin(-\theta) = -\sin(\theta) \)

These identities indicate that the cosine function is an even function, meaning it does not change with negative angles, while the sine function is odd, changing the sign when the angle is negative.

These properties are more than just mathematical quirks—they have practical applications. Understanding them can help in analyzing wave functions, alternating currents in circuit analysis, and more. By mastering trigonometric identities, you'll have a much richer toolset for exploring the world of complex numbers.