Problem 62 - A Graphical Approach to Precalculus with Limits

Question

Find each product in rectangular form, using exact values. $$\left[6 \operatorname{cis} \frac{2 \pi}{3}\right]\left[5 \operatorname{cis}\left(-\frac{\pi}{6}\right)\right]$$

Step-by-Step Solution

Answer

The product is \(30i\) in rectangular form.

Step 1: Understand the Problem

We need to multiply two complex numbers given in polar form: \(6 \operatorname{cis} \frac{2\pi}{3}\) and \(5 \operatorname{cis} \left(-\frac{\pi}{6}\right)\). The operation involves using the formula: \(r_1 \operatorname{cis} \theta_1 \cdot r_2 \operatorname{cis} \theta_2 = r_1 r_2 \operatorname{cis}(\theta_1 + \theta_2)\).

Step 2: Multiply the Moduli

Multiply the moduli (magnitudes) of the two complex numbers:\[6 \times 5 = 30\]

Step 3: Add the Arguments

Add the arguments (angles) of the two complex numbers:\[\frac{2\pi}{3} + \left(-\frac{\pi}{6}\right) = \frac{2\pi}{3} - \frac{\pi}{6}\] To add the fractions, find a common denominator:\[\frac{2\pi}{3} = \frac{4\pi}{6}\] and \[-\frac{\pi}{6} = -\frac{\pi}{6}\]Thus, \[\frac{4\pi}{6} - \frac{\pi}{6} = \frac{3\pi}{6} = \frac{\pi}{2}\].

Step 4: Write the Result in Polar Form

The product in polar form is \[30 \operatorname{cis} \frac{\pi}{2}\].

Step 5: Convert the Result to Rectangular Form

Convert the polar form back to rectangular form using \(30(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2})\).We know \(\cos \frac{\pi}{2} = 0\) and \(\sin \frac{\pi}{2} = 1\). So:\[30(0 + i \cdot 1) = 30i\].

Key Concepts

Polar Form

When dealing with complex numbers, the polar form is a useful way to express them, especially when it involves multiplication or division. In the polar form, a complex number is represented as \( r \operatorname{cis} \theta \), where \( r \) is the modulus (or magnitude) and \( \theta \) is the argument (or angle). This form leverages trigonometric functions to relate complex numbers to the unit circle on the complex plane. The formula \( r(\cos \theta + i\sin \theta) \) is often abbreviated as \( r \operatorname{cis} \theta \), making it compact and easy to manipulate.

The key benefit of using polar form arises in operations like multiplication. When multiplying complex numbers in polar form, you multiply their moduli and add their arguments. This property simplifies computations significantly compared to the rectangular form. For example, multiplying two complex numbers, \( 6 \operatorname{cis} \frac{2\pi}{3} \) and \( 5 \operatorname{cis} \left(-\frac{\pi}{6}\right) \), gives you \( 30 \operatorname{cis} \frac{\pi}{2} \) after performing these straightforward calculations.

This method highlights the elegance and efficiency of polar form in handling such tasks, especially in comparison to the more cumbersome process in rectangular form.

Rectangular Form

The rectangular form of a complex number is probably the form that most people are familiar with. It takes the format \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part of the complex number. This form directly relates to the Cartesian coordinate system, where complex numbers are plotted as points. The real part \( a \) represents the horizontal distance from the origin, while \( b \), the imaginary part, represents the vertical distance.

Converting a complex number from polar to rectangular form involves using trigonometry. For a complex number in polar form \( r \operatorname{cis} \theta \), its rectangular form is found using the formulas \( a = r \cos \theta \) and \( b = r \sin \theta \), giving the equation \( a + bi \).

As in our problem, the polar form \( 30 \operatorname{cis} \frac{\pi}{2} \) translates to the rectangular form \( 30i \) since \( \cos \frac{\pi}{2} = 0 \) and \( \sin \frac{\pi}{2} = 1 \), resulting in \( 30(0 + i \cdot 1) = 30i \). This exercise showcases why understanding both forms is crucial, as it helps in converting and interpreting complex numbers efficiently.

Cis Notation

Cis notation is a streamlined way to express complex numbers, especially in polar form. The notation \( \operatorname{cis} \theta \) stands for \( \cos \theta + i \sin \theta \). This abbreviation simplifies writing and calculations involving trigonometric forms of complex numbers.

By using \( \operatorname{cis} \), we reduce the complexity and potential for errors when managing the sine and cosine components separately. In operations like multiplication and division, this notation shines by allowing concise expression and manipulation of complex numbers. For instance, \( r_1 \operatorname{cis} \theta_1 \cdot r_2 \operatorname{cis} \theta_2 = r_1r_2 \operatorname{cis}(\theta_1 + \theta_2) \) highlights its power.

The utility of \( \operatorname{cis} \) becomes evident when you multiply complex numbers with ease. The common operations, such as seen in our original exercise, involve direct algebraic transformations that \( \operatorname{cis} \) notation facilitates. It bridges the succinctness necessary for complex arithmetic with the robustness required for precise mathematical articulation.