Problem 82 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Without actually performing the operations, state why the products $$\left[2\left(\cos 45^{\circ}+i \sin 45^{\circ}\right)\right]\left[5\left(\cos 90^{\circ}+i \sin 90^{\circ}\right)\right]$$ and $$\begin{array}{r}\left[2\left(\cos \left(-315^{\circ}\right)+i \sin \left(-315^{\circ}\right)\right)\right] \\ \left[5\left(\cos \left(-270^{\circ}\right)+i \sin \left(-270^{\circ}\right)\right)\right]\end{array}$$ are the same.

Step-by-Step Solution

Answer

The products are equal because both have the same magnitude (10) and equivalent angles (135°).

Step 1: Understanding the problem

Both expressions represent products of complex numbers in polar form. The task requires identifying why these products are equal without direct calculation.

Step 2: Examine the structure of complex products

In polar form, a complex number is expressed as \( r(\cos \theta + i \sin \theta) \), where \( r \) is the magnitude and \( \theta \) is the argument (angle). The product of two complex numbers is found by multiplying their magnitudes and adding their arguments.

Step 3: Identify magnitudes in both expressions

For both products, the magnitudes are \( 2 \cdot 5 = 10 \).

Step 4: Calculate arguments

In the first product, the angles are \( 45^\circ \) and \( 90^\circ \). The sum is \( 135^\circ \). In the second product, the angles are \( -315^\circ \) and \( -270^\circ \). The sum is \( -315^\circ + (-270^\circ) = -585^\circ \).

Step 5: Compare arguments using equivalent angles

Angles are equivalent if they differ by a multiple of \(360^\circ\). The angle \(-585^\circ\) is equivalent to \( -585^\circ + 720^\circ = 135^\circ \).

Step 6: Conclusion

Since both expressions result in the same magnitude \(10\) and equivalent angle \(135^\circ\), the products are indeed the same.

Key Concepts

Understanding Polar Form of Complex Numbers

Complex numbers can seem intimidating, but expressing them in polar form can make operations like multiplication much simpler. A complex number in polar form is written as \( r(\cos \theta + i \sin \theta) \). Here, \( r \) represents the magnitude (or length) of the vector representing the complex number, while \( \theta \) is the angle (or argument) the vector makes with the positive real axis. This form is particularly useful for multiplication, as it involves straightforward operations on \( r \) and \( \theta \) rather than on complex numbers directly. This expression highlights how powerful polar form can be in managing complex multiplication efficiently.

• \( r \) is the distance from the origin to the point in the complex plane.
• \( \theta \) is measured from the positive x-axis counter-clockwise.
• This form simplifies operations like multiplication, as you'll see next.

Octonion provides an elegant way to view and manipulate complex numbers geometrically.

Magnitude and Argument in Complex Numbers

Understanding magnitude and argument is crucial for working with polar representation of complex numbers. The magnitude, \( r \), is a positive real number representing the distance from the origin to the complex number in the plane. The argument, \( \theta \), is the angle from the positive real axis. Both concepts are key when performing operations like multiplication.To find the magnitude and argument:

• Magnitudes in the original exercise are calculated by multiplying the single magnitudes: \( 2 \times 5 = 10 \).
• Arguments are summed: In the first product, \( 45^\circ + 90^\circ = 135^\circ \); In the second, \(-315^\circ + (-270^\circ) = -585^\circ \).

These values provide insight into the behavior and equivalence of the complex products, as further discussed with equivalent angles.

Complex Multiplication Using Polar Form

Multiplying complex numbers in polar form involves a neat process of multiplying their magnitudes and adding their angles. This method is simple yet profound, as it turns an otherwise complicated arithmetic task into one that's geometrically intuitive.

• The magnitudes are multiplied. For example, in our exercise, both products have magnitudes \( 2 \times 5 = 10 \).
• The angles or arguments are added. This step is crucial for understanding why the products are equal despite appearing different initially.

These properties show why polar form is extremely helpful in simplifying complex operations, as it aligns with geometric transformations in the complex plane.

Understanding Equivalent Angles

Angles in the context of polar coordinates have a peculiar property of being cyclical. Equivalent angles come into play when angles that differ by multiples of \(360^\circ\) represent the same direction in the plane.
In the exercise, while the angles \( 135^\circ \) and \(-585^\circ \) may look distinct, they indeed describe the same line in the complex plane.

• An angle is turned equivalent by adding or subtracting \(360^\circ\) until it fits within a \(0^\circ\) to \(360^\circ\) range.
• For example, \(-585^\circ + 720^\circ = 135^\circ\).

This cyclical nature of angles proofs the products result in the same complex number, hence validating the equivalence in outcomes as shown in polar form's utility.