Problem 61 - A Graphical Approach to Precalculus with Limits

Question

Find each product in rectangular form, using exact values. $$\left[5 \text { cis } \frac{\pi}{2}\right]\left[3 \text { cis } \frac{\pi}{4}\right]$$

Step-by-Step Solution

Answer

The product is \(-\frac{15\sqrt{2}}{2} + i\frac{15\sqrt{2}}{2}\)."

Step 1: Understand the Problem

We need to find the product of two complex numbers given in polar form, where each number is in the format \( r \text{ cis } \theta \), which is shorthand for \( r(\cos \theta + i \sin \theta) \).

Step 2: Find the Magnitude of the Product

We calculate the product of the magnitudes (radii) of the two complex numbers. The magnitudes are 5 and 3 respectively, so the magnitude of the product is \( 5 \times 3 = 15 \).

Step 3: Find the Argument of the Product

Add the angles \(\theta\) of the two complex numbers: \( \frac{\pi}{2} + \frac{\pi}{4} = \frac{2\pi}{4} + \frac{\pi}{4} = \frac{3\pi}{4} \). This is the argument of the product.

Step 4: Convert to Rectangular Form

Use the polar to rectangular form conversion: \( r \text{ cis } \theta = r(\cos \theta + i \sin \theta) \). Here, substitute \( r = 15 \) and \( \theta = \frac{3\pi}{4} \):- \(\cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2}\)- \(\sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2}\)Thus, the rectangular form is \( 15(-\frac{\sqrt{2}}{2}) + 15i(\frac{\sqrt{2}}{2}) = -\frac{15\sqrt{2}}{2} + i\frac{15\sqrt{2}}{2} \).

Key Concepts

Polar Form

Complex numbers can be represented using the polar form because it provides a intuitive way of understanding them in terms of rotation and magnitude. The polar form of a complex number is expressed as \( r \text{ cis } \theta \), where \( r \) is the magnitude or modulus of the complex number, and \( \theta \) is the angle, also known as the argument, which the line representing the complex number makes with the positive real axis.

To convert a complex number from polar to standard form, you use trigonometric functions: \( r \text{ cis } \theta = r(\cos \theta + i \sin \theta) \). This 'cis' notation is shorthand and useful when dealing with multiple complex numbers, especially when performing multiplication and division.

Using this form simplifies multiplication since you can multiply the magnitudes and add the angles. This makes solving problems like the one in the original exercise more straightforward.

Rectangular Form

Rectangular form is another way to express complex numbers, and it is simply the sum of the real part and the imaginary part. It uses the format \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part.

When converting from polar to rectangular form, you use the equations \( x = r\cos \theta \) and \( y = r\sin \theta \), where \( x \) is the real component and \( y \) the imaginary component of the complex number. For example, the complex number \( 15 \text{ cis } \frac{3\pi}{4} \) converts to \(-\frac{15\sqrt{2}}{2} + i\frac{15\sqrt{2}}{2} \) in rectangular form, as calculated from the respective cosine and sine values.

Understanding both forms and how to convert between them is crucial in complex number arithmetic, helping you fully solve exercises with precision and clarity.

Magnitude and Argument

In the context of polar form, the magnitude and argument are fundamental concepts. The magnitude, often denoted as \( |z| \), is the distance from the origin to the point representing the complex number on the complex plane, calculated as \( r = \sqrt{x^2 + y^2} \) when in rectangular form.

• Magnitude: Gives the length or absolute size of the complex number.
• Argument: Measured in radians or degrees, represents the angle formed with the positive direction of the real axis.

In polar form, tasks like multiplication become more manageable; you multiply magnitudes and add arguments: \( |z| = 5 \times 3 = 15 \) and \( \theta = \frac{\pi}{2} + \frac{\pi}{4} = \frac{3\pi}{4} \). This is the polar form formula applied step-by-step to solve the exercise effectively.

Trigonometric Functions

Trigonometric functions are integral for converting complex numbers between forms. They are used to determine the coordinates of a point on the unit circle from an angle. The functions \( \cos \theta \) and \( \sin \theta \) provide the real and imaginary parts, respectively, in the conversion from polar to rectangular form.

For example, \( \cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2} \) and \( \sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2} \). These values help to express the rectangular form as \( -\frac{15\sqrt{2}}{2} + i\frac{15\sqrt{2}}{2} \).

These functions are not only essential for conversions but also for understanding complex number operations like addition of arguments and rotations on the complex plane. Mastery of these trigonometric forms and functions enables solving any complex number problem accurately and efficiently.