Problem 53 - Contemporary Precalculus

Question

In Exercises \(53-64,\) perform the indicated multiplication or division. Express your answer in both polar form \(r(\cos \theta+i \sin \theta)\) and rectangular form \(a+b i\). $$\left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right) \cdot(\cos \pi+i \sin \pi)$$

Step-by-Step Solution

Answer

Question: Multiply the complex numbers \(\left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right)\) and \((\cos \pi+i \sin \pi)\) and express the result in both polar and rectangular forms. Answer: Polar form: \(1(\cos \frac{3\pi}{2} + i \sin \frac{3\pi}{2})\), Rectangular form: \(0 - 1i\)

Step 1: Recall the formula for the product of two complex numbers in polar form

Given two complex numbers in polar form, \(r_1(\cos \theta_1 + i \sin \theta_1)\) and \(r_2(\cos \theta_2 + i \sin \theta_2)\), their product can be found using the formula: $$(r_1 \cdot r_2)(\cos (\theta_1 + \theta_2) + i \sin (\theta_1 + \theta_2))$$ In our case, \(r_1 = 1, \theta_1 = \frac{\pi}{2}, r_2 = 1,\) and \(\theta_2 = \pi\).

Step 2: Compute the magnitude and angle of the product

First, compute the magnitude and angle of the product: $$r_1 \cdot r_2 = 1 \cdot 1 = 1$$ $$\theta_1 + \theta_2 = \frac{\pi}{2} + \pi = \frac{3\pi}{2}$$

Step 3: Write the product in polar form

Write the product using the magnitude and angle found above: $$1(\cos \frac{3\pi}{2} + i \sin \frac{3\pi}{2})$$

Step 4: Convert to rectangular form

Now, we need to convert the product to rectangular form. In general, to convert a complex number from polar form to rectangular form, we can use the following formulas: $$a = r \cos \theta$$ $$b = r \sin \theta$$ In our case \(r = 1\) and \(\theta = \frac{3\pi}{2}\), so calculate \(a\) and \(b\): $$a = 1 \cdot \cos \frac{3\pi}{2} = 0$$ $$b = 1 \cdot \sin \frac{3\pi}{2} = -1$$

Step 5: Write the final results

The product is now expressed in both polar and rectangular form: Polar form: $$1(\cos \frac{3\pi}{2} + i \sin \frac{3\pi}{2})$$ Rectangular form: $$0 - 1i$$

Key Concepts

Polar Form of Complex Numbers

Understanding the polar form of complex numbers is essential for simplifying the multiplication and division of complex quantities. Unlike the familiar rectangular form, which uses a standard x-y coordinate system to represent complex numbers as points, the polar form relies on two different pieces of information: a magnitude and an angle.

In polar form, the magnitude, often denoted as \( r \), represents the distance from the origin to the point in the complex plane, while the angle \( \theta \), typically measured in radians, represents the direction of the line connecting the origin to the point. The polar form is written as \( r(\cos \theta + i \sin \theta) \), where \( i \) is the imaginary unit. This representation can make complex number operations more intuitive, particularly when multiplying or dividing, as it reflects the number's properties in terms of its size and rotation from the real axis.

One practical aspect is that the polar form is especially conducive to multiplying complex numbers since it simplifies the process to multiplying their magnitudes and adding their angles, an invaluable trait when dealing with trigonometric identities or transformations between different coordinate systems.

Rectangular Form of Complex Numbers

The rectangular form of a complex number is perhaps the most familiar representation. Here, a complex number is expressed as \( a + bi \), where \( a \) and \( b \) are real numbers that correspond to the x and y coordinates, respectively, on the complex plane, with \( i \) again standing for the imaginary unit.

This form of representation makes use of the Cartesian coordinate grid structure and allows for straightforward addition and subtraction of complex numbers, as one would do with two-dimensional vectors. Each part, the real component \( a \) and the imaginary component \( bi \), can be manipulated separately, making it analogous to working with horizontal and vertical components in physics.

Converting between rectangular form and other representations of complex numbers, such as polar form, involves trigonometry and can provide a deeper insight into the number's properties, such as its absolute value being the distance from the origin, and its argument being the angle it makes with the real axis.

Complex Numbers Multiplication

Complex numbers multiplication can be intuitively handled by considering the geometric interpretation of complex numbers, especially when they are in polar form. For instance, let's consider the product of two complex numbers in polar form: \( r_1(\cos \theta_1 + i \sin \theta_1) \) and \( r_2(\cos \theta_2 + i \sin \theta_2) \).

The multiplication of these two numbers entails multiplying their magnitudes \( r_1 \) and \( r_2 \), and adding their angles \( \theta_1 \) and \( \theta_2 \), resulting in a new complex number with a magnitude of \( r_1 \times r_2 \) and an angle of \( \theta_1 + \theta_2 \). This results from the trigonometric addition formulas, simplifying the multiplication to \( (r_1 \cdot r_2)(\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)) \), which beautifully demonstrates the complex multiplication rule that combines magnitudes and sums angles.

Compared to rectangular form, where multiplying would necessitate distribution over addition and applying the rule that \( i^2 = -1 \), polar form can significantly streamline complex multiplication, particularly useful in fields such as electrical engineering and quantum mechanics where complex numbers frequently emerge.

Converting Polar to Rectangular Form

When working with complex numbers, it's often necessary to switch between polar and rectangular forms, depending on the context and the operations involved. To convert a complex number from polar to rectangular form, knowledge of basic trigonometric functions is essential.

The conversion process makes use of the formulas \( a = r \cos \theta \) and \( b = r \sin \theta \), where \( a \) and \( b \) are the real and imaginary components of the rectangular form, respectively. This method effectively translates the magnitude and angle information of the polar form into the horizontal and vertical displacements in the Cartesian plane of the rectangular form.

Such conversion can be crucial when required to perform operations that are more straightforward in one form or the other, such as addition and subtraction being easier in rectangular form, while multiplication and division are simplified in polar form. Enabling students to switch between these forms in a step-by-step manner bolsters their understanding of complex numbers and expands the tools at their disposal for tackling diverse mathematical challenges.