Problem 38 - Algebra and Trigonometry

Question

Find \(z w\) and \(\frac{z}{w} .\) Write each answer in polar form and in exponential form. \(z=\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\) \(w=\cos \frac{5 \pi}{9}+i \sin \frac{5 \pi}{9}\)

Step-by-Step Solution

Answer

The product is \(zw = e^{i \frac{11\pi}{9}} = \text{cis}(\frac{11\pi}{9})\). The quotient is \(\frac{z}{w} = e^{i \frac{\pi}{9}} = \text{cis}(\frac{\pi}{9})\).

Step 1 - Write both numbers in exponential form

The complex number in polar form is written as:\[ z = \text{cis}(\theta_z) = \text{cis} \left(\frac{2\pi}{3}\right) \]\[ w = \text{cis}(\theta_w) = \text{cis} \left(\frac{5\pi}{9}\right)\]In exponential form, they are:\[ z = e^{i \frac{2\pi}{3}} \]\[ w = e^{i \frac{5\pi}{9}} \]

Step 2 - Find the product \(zw\) in exponential form

Using the property of exponents for multiplication, \(e^{i\theta_1} e^{i\theta_2} = e^{i(\theta_1 + \theta_2)}\):\[ z w = e^{i \frac{2\pi}{3}} \cdot e^{i \frac{5\pi}{9}} = e^{i \left(\frac{2\pi}{3} + \frac{5\pi}{9}\right)} \]

Step 3 - Simplify the angle for the product

Combine the angles in the exponent:\[ \frac{2\pi}{3} + \frac{5\pi}{9} = \frac{6\pi}{9} + \frac{5\pi}{9} = \frac{11\pi}{9} \]Thus, \[ zw = e^{i \frac{11\pi}{9}} \]

Step 4 - Express \(zw\) in polar form

Since \(e^{i\theta} = \text{cis}(\theta)\), we get:\[ zw = \text{cis} \left(\frac{11\pi}{9}\right) \]

Step 5 - Find the quotient \(\frac{z}{w}\) in exponential form

Using the property for division of exponents, \(\frac{e^{i\theta_1}}{e^{i\theta_2}} = e^{i(\theta_1 - \theta_2)}\):\[ \frac{z}{w} = \frac{e^{i \frac{2\pi}{3}}}{e^{i \frac{5\pi}{9}}} = e^{i \left(\frac{2\pi}{3} - \frac{5\pi}{9}\right)} \]

Step 6 - Simplify the angle for the quotient

Combine the angles in the exponent:\[ \frac{2\pi}{3} - \frac{5\pi}{9} = \frac{6\pi}{9} - \frac{5\pi}{9} = \frac{\pi}{9} \]Thus, \[ \frac{z}{w} = e^{i \frac{\pi}{9}} \]

Step 7 - Express \(\frac{z}{w}\) in polar form

Since \(e^{i\theta} = \text{cis}(\theta)\), we have:\[ \frac{z}{w} = \text{cis} \left(\frac{\pi}{9}\right) \]

Key Concepts

polar form

Polar form is a way of expressing complex numbers using a combination of a magnitude (or modulus) and an angle (or argument). This is very useful for understanding the geometric representation of complex numbers on the complex plane. A complex number in polar form is represented as:
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\( z = r \text{cis}(\theta) \), where \( r \) is the magnitude and \( \theta \) is the argument.
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The \( \text{cis} \) means \( \text{cos} + i\text{sin} \). For example, \( z = \text{cos} \frac{2 \pi}{3} + i \text{sin} \frac{2 \pi}{3} \) can be written in polar form as:
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\( z = \text{cis}(\frac{2 \pi}{3}) \). This form helps us easily identify the angle and modulus of the complex number.

exponential form

The exponential form provides another way to represent complex numbers, taking advantage of Euler's formula, which states that \( e^{i\theta} = \text{cos}(\theta) + i\text{sin}(\theta) \). Converting a complex number into exponential form can simplify many operations like multiplication and division.
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Using the previous example, \( z = \text{cis}(\frac{2 \pi}{3}) \) can be written in exponential form as:
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\( z = e^{i \frac{2 \pi}{3}} \). Similarly, for \( w = \text{cis}(\frac{5 \pi}{9}) \), it becomes:
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\( w = e^{i \frac{5 \pi}{9}} \). This representation makes it straightforward to use the properties of exponents to perform operations on complex numbers.

multiplication of complex numbers

Multiplying complex numbers in polar or exponential form is much simpler compared to rectangular form. The rule for multiplying them is to multiply their magnitudes and add their arguments. For instance, given:
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\( z = e^{i \frac{2 \pi}{3}} \) and \( w = e^{i \frac{5 \pi}{9}} \), the product is:
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\( z w = e^{i \frac{2 \pi}{3}} \times e^{i \frac{5 \pi}{9}} = e^{i \big(\frac{2 \pi}{3} + \frac{5 \pi}{9}\big)} \)
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Simplifying the angle we get:
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\( \frac{2 \pi}{3} + \frac{5 \pi}{9} = \frac{6 \pi}{9} + \frac{5 \pi}{9} = \frac{11 \pi}{9} \). Thus:
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\( z w = e^{i \frac{11 \pi}{9}} \), or \( z w = \text{cis} \big( \frac{11 \pi}{9} \big) \).

division of complex numbers

Dividing complex numbers in polar or exponential form is also straightforward. The rule is to divide their magnitudes and subtract their arguments. Let's take the same two complex numbers:
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\( z = e^{i \frac{2 \pi}{3}} \) and \( w = e^{i \frac{5 \pi}{9}} \). The quotient is:
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\( \frac{z}{w} = \frac{e^{i \frac{2 \pi}{3}}}{e^{i \frac{5 \pi}{9}}} = e^{i \big(\frac{2 \pi}{3} - \frac{5 \pi}{9}\big)} \). Simplifying the angle we get:
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\( \frac{2 \pi}{3} - \frac{5 \pi}{9} = \frac{6 \pi}{9} - \frac{5 \pi}{9} = \frac{ \pi}{9} \). Thus:
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\( \frac{z}{w} = e^{i \frac{\pi}{9}} \), or \( \frac{z}{w} = \text{cis} \big( \frac{\pi}{9} \big) \). This method simplifies the division process into basic algebraic addition and subtraction of angles.