Problem 69 - A Graphical Approach to Precalculus with Limits

Question

Find each product in rectangular form, using exact values. $$\frac{3 \operatorname{cis}\left(\frac{61 \pi}{36}\right)}{9 \operatorname{cis}\left(\frac{13 \pi}{36}\right)}$$

Step-by-Step Solution

Answer

The product in rectangular form is \( -\frac{1}{6} - \frac{\sqrt{3}}{6} i \).

Step 1: Convert Expression to Simplified Polar Form

The given expression \( \frac{3 \operatorname{cis}\left(\frac{61 \pi}{36}\right)}{9 \operatorname{cis}\left(\frac{13 \pi}{36}\right)} \) can be simplified by dividing the magnitudes and subtracting the angles. The magnitude becomes \( \frac{3}{9} = \frac{1}{3} \) and the angle simplifies as \( \frac{61 \pi}{36} - \frac{13 \pi}{36} = \frac{48 \pi}{36} = \frac{4 \pi}{3} \). Thus, the polar form is \( \frac{1}{3} \operatorname{cis}\left(\frac{4\pi}{3}\right) \).

Step 2: Convert Polar Form to Rectangular Form

The polar form \( \frac{1}{3} \operatorname{cis}\left(\frac{4\pi}{3}\right) \) must be converted to rectangular form. Using the formula \( r \operatorname{cis}(\theta) = r(\cos(\theta) + i\sin(\theta)) \), we find \( \cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2} \) and \( \sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2} \). Therefore, the rectangular form is \( \frac{1}{3} \left(-\frac{1}{2} - \frac{\sqrt{3}}{2} i \right) \).

Step 3: Simplify the Rectangular Coordinates

Now simplify the expression \( \frac{1}{3} \left(-\frac{1}{2} - \frac{\sqrt{3}}{2} i \right) \) by distributing \( \frac{1}{3} \). This results in \( -\frac{1}{6} - \frac{\sqrt{3}}{6} i \). Thus, the final rectangular form is \( -\frac{1}{6} - \frac{\sqrt{3}}{6} i \).

Key Concepts

Cis Function

The cis function is an important tool when working with complex numbers in polar form. It abbreviates the expression involving cosine and sine functions, typically written as \( \operatorname{cis}(\theta) \), where \( \theta \) is an angle measured in radians. Specifically, \( \operatorname{cis}(\theta) = \cos(\theta) + i\sin(\theta) \). This expression effectively represents a complex number on the unit circle using the angle \( \theta \).

• Applications: The cis function is particularly useful in simplifying multiplication and division of complex numbers. This is because, in polar form, adding angles represents multiplication and subtracting represents division.
• Why Use Cis: Writing complex numbers as \( r \operatorname{cis}(\theta) \), where \( r \) is the magnitude, allows for straightforward manipulation. The magnitude reflects the distance from the origin, while \( \theta \) represents the direction.

For example, to multiply two complex numbers in polar form, you just multiply their magnitudes and add their angles. This simplicity is why the cis function is preferred for operations in polar form.

Complex Numbers

Complex numbers are a combination of real and imaginary numbers, expressed in the form \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part. The imaginary unit \( i \) is defined as \( \sqrt{-1} \).

• Rectangular Form: This is the conventional form, \( a + bi \), used to plot complex numbers on a Cartesian plane.
• Polar Form: Alternatively, complex numbers can be represented as \( r \operatorname{cis}(\theta) \). Here, \( r \) is the magnitude \( \sqrt{a^2 + b^2} \), and \( \theta \) is the angle formed with the positive real axis.

The polar form is particularly advantageous in simplifying multiplication and division of complex numbers. This is because angle operations (addition or subtraction) are easier than their counterparts in rectangular form.

Exact Values

Exact values refer to the trigonometric values that are derived without approximation, often involving special angles like \( \frac{\pi}{3} \), \( \frac{\pi}{4} \), and \( \frac{\pi}{6} \). These angles have well-known sine and cosine values that are essential for converting between forms.

• Importance in Conversion: When converting from polar to rectangular form, exact trigonometric values ensure precision. For instance, \( \cos\left(\frac{4\pi}{3}\right) \) and \( \sin\left(\frac{4\pi}{3}\right) \) are exact values used in transformations.
• Common Exact Values: Some common exact values include \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \), \( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \), and their respective transformations for other quadrant angles.

Using these values helps in maintaining accuracy throughout complex arithmetic calculations, particularly when converting between the rectangular and polar forms of complex numbers.