Problem 32 - Calculus

Question

The polar form of \(r e^{i \theta}\) times \(R e^{4}\) is ______ . The rectangular form is _________. Circle the terms that give \(r R \cos (\theta+\phi)\)

Step-by-Step Solution

Answer

The polar form is \(rR e^{i(\theta+4)}\). The rectangular form is \(rR \cos(\theta+4) + i rR \sin(\theta+4)\). Circle \(rR \cos(\theta+4)\).

Step 1: Identify the Polar Form Components

The given expressions are in polar form: \(r e^{i \theta}\) and \(R e^4\). Identify these components: \(r\) is the magnitude and \(\theta\) is the angle for the first expression, while \(R\) is the magnitude for the second expression and it does not have an imaginary part.

Step 2: Multiply Magnitudes and Add Angles

Multiply the magnitudes \(r\) and \(R\) together. Since the second expression \(R e^4\) does not have an angle in polar form, treat \(e^4\) as a constant, essentially \(\cos(0) + i\sin(0)\). Therefore, the polar form is \(rR e^{i \theta}e^4 = rR e^{i \theta + 4}\).

Step 3: Convert Polar to Rectangular Form

To convert \(rR e^{i(\theta + 4)}\) to rectangular form, use Euler's formula: \(x + iy = rR(\cos(\theta + 4) + i\sin(\theta + 4))\). This gives the rectangular form as: \(x = rR \cos(\theta + 4)\) and \(y = rR \sin(\theta + 4)\).

Step 4: Circle Terms Matching Desired Expression

Identify the terms from the rectangular form that match the structure \(r R \cos(\theta + \phi)\). This term is \(x = rR \cos(\theta + 4)\), which matches the expression structure when we assume \(\phi = 4\).

Key Concepts

Polar Form

Polar form represents complex numbers using a combination of a magnitude and an angle. Imagine a point in the 2-dimensional plane with a specific distance from the origin. This distance is known as the magnitude, denoted as \( r \).
The angle, often denoted as \( \theta \), is the direction of the line from the origin to the point relative to the positive x-axis.

In polar form, a complex number is expressed as \( re^{i \theta} \), where:

• \( r \) is the magnitude
• \( e^{i \theta} \) incorporates the angle \( \theta \)

This form is highly efficient when multiplying complex numbers, as it requires only multiplying the magnitudes and adding the angles.

Rectangular Form

Rectangular Form refers to the standard representation of complex numbers on a plane using x and y coordinates. In mathematical terms, a complex number \( z \) is written as \( x + iy \).
Here, the real part \( x \) corresponds to the x-coordinate, and the imaginary part \( y \) is the y-coordinate scaled by the imaginary unit \( i \).

This form is very intuitive as it directly shows the complex number's location on a Cartesian plane. Converting between rectangular and polar forms can be done using trigonometric relationships, involving \( \cos \) and \( \sin \) for conversion to rectangular form.

Euler's Formula

Euler's formula elegantly bridges polar and rectangular forms of complex numbers. It states that for any real number \( \theta \), the complex number \( e^{i\theta} = \cos(\theta) + i\sin(\theta) \).

This amazing formula helps in converting polar form expressions to their rectangular equivalent. By plugging into Euler's formula, you replace \( e^{i\theta} \) with trigonometric terms, transforming polar expressions into the familiar \( x + iy \) format.

• Useful for calculations in engineering, physics.
• Simplifies multiplication and exponentiation of complex numbers.

This versatility showcases why Euler's formula is considered one of the most beautiful equations in mathematics.

Magnitude and Angle

The magnitude and angle are fundamental aspects of a complex number when represented in polar form.

The magnitude \( r \) of a complex number \( a + bi \) is calculated as \( \sqrt{a^2 + b^2} \), giving the length of the vector from the origin to the point \( (a, b) \).

The angle \( \theta \) describes the orientation of that vector and is found using the arctangent function \( \theta = \tan^{-1}( \frac{b}{a} ) \). These concepts allow one to fully capture both the size and direction of a complex number, making operations like multiplication more intuitive, as these can be handled through simple geometry like scaling and rotation.