Problem 108 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The proof of the formula for the product of two complex numbers in polar form uses the sum formulas for cosines and sines that I studied in the previous chapter.

Step-by-Step Solution

Answer

The statement partially makes sense as the sum formulas for cosines and sines are relevant in dealing with operations on complex numbers, but they are not directly used to derive the multiplication rule for complex numbers in polar form. This rule comes more directly from the operations on complex numbers.

Step 1: Understanding the statement

First, make sense of the statement given. The formula for the product of two complex numbers in polar form is \(r_1 r_2 (cos(\theta_1 + \theta_2) + i sin(\theta_1 + \theta_2))\), where \(r_1, r_2\) are the magnitudes and \(\theta_1, \theta_2\) are the angles of the two complex numbers respectively. The statement asserts that this formula is derived using the sum formulas for cosines and sines.

Step 2: Evaluate the statement

Next, recall the sum identities: \(cos(a + b) = cos(a)cos(b) - sin(a)sin(b)\) and \(sin(a + b) = sin(a)cos(b) + cos(a)sin(b)\). These identities do not directly relate to deriving the formula for the product of two complex numbers in polar form. However, the addition within the cosine and sine functions in the formula aligns with these sum identities, implying a certain level of relevance. Therefore, the statement partially makes sense.

Step 3: Final reasoning

However, while the sum identities for cosines and sines are useful in dealing with complex numbers, they are not directly utilized in deriving the multiplication rule for complex numbers in polar form. This rule comes from operations from complex numbers, where multiplication is done by adding the angles and multiplying the magnitudes. Therefore, it does not completely make sense to say that the formula derives from those identities.

Key Concepts

Polar Form

Complex numbers can be represented in different ways, one of which is the Polar form. This form is especially useful for multiplication and other operations. A complex number in polar form is expressed as:

• The magnitude ( \( r \)) represents the distance from the origin to the point on the complex plane.


• The angle ( \( \theta \)) shows the direction of the vector relative to the positive x-axis.

In mathematical terms, polar form is expressed as:\[r(\cos \theta + i \sin \theta)\]Where \( r \) and \( \theta \) are the magnitude and angle respectively. Using Euler's formula, it can also be written as:\[r e^{i\theta}\]This form makes it easier to perform operations like multiplication and division, due to the properties of exponents and angles.

Multiplication of Complex Numbers

Multiplying complex numbers is simplified when they are in polar form. This is because of the geometric interpretation of multiplication in this form.When we multiply two complex numbers, say \( z_1 = r_1(\cos \theta_1 + i \sin \theta_1) \) and \( z_2 = r_2(\cos \theta_2 + i \sin \theta_2) \), the result is:\[z_1 \cdot z_2 = r_1r_2(\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2))\]Here's what happens step-by-step:

• The magnitudes \( r_1 \) and \( r_2 \) are multiplied.
• The angles \( \theta_1 \) and \( \theta_2 \) are added.

This means the multiplication of two complex numbers involves multiplying their magnitudes and adding their angles. Doing the multiplication in rectangular form is cumbersome compared to the polar form's elegance and simplicity.

Trigonometric Identities

Trigonometric identities are valuable tools in mathematics. They become handy in many calculations, especially involving angles and rotations.The sum formulas for cosine and sine are particularly worth remembering:

• \( \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \)
• \( \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \)

Though these identities don't directly derive the multiplication formula of complex numbers in polar form, they explain why the addition of angles in the formula aligns with our understanding of trigonometry.
They are fundamental within trigonometric calculations and also frequently appear in complex numbers scenarios, aiding in various transformations and simplifications.