Problem 74 - Precalculus

Question

Determine whether the statement is true or false. Let \(z_{1}=r_{1}\left(\cos \theta_{1}+i \sin \theta_{1}\right)=r_{1} e^{j \theta_{1}}\) and \(z_{2}=r_{2}\left(\cos \theta_{2}+i \sin \theta_{2}\right)=r_{2} e^{i \theta,}\) be two complex numbers. Use the properties of exponentials to show that \(\frac{z_{1}}{z_{2}}=\frac{r_{1}}{r_{2}}\left[\cos \left(\theta_{1}-\theta_{2}\right)+i \sin \left(\theta_{1}-\theta_{2}\right)\right]\).

Step-by-Step Solution

Answer

The statement is true.

Step 1: Express Each Complex Number Using Exponential Form

We begin by expressing the complex numbers \(z_1\) and \(z_2\) in their exponential forms. For \(z_1\), we have \(z_1 = r_1 e^{i \theta_1}\). Similarly, for \(z_2\), we have \(z_2 = r_2 e^{i \theta_2}\). This form takes advantage of Euler's formula, which relates trigonometric functions to exponential functions.

Step 2: Form the Quotient of the Two Complex Numbers

To find \(\frac{z_1}{z_2}\), we divide \(z_1\) by \(z_2\):\[\frac{z_1}{z_2} = \frac{r_1 e^{i \theta_1}}{r_2 e^{i \theta_2}}.\]

Step 3: Simplify the Magnitude

The magnitudes \(r_1\) and \(r_2\) are real numbers, so the fraction of these magnitudes simplifies to:\[\frac{r_1}{r_2}.\]

Step 4: Apply the Exponential Property of Quotients

Using the property of exponentials \( \frac{e^{a}}{e^{b}} = e^{a-b} \), we can simplify the exponential part:\[e^{i \theta_1 - i \theta_2} = e^{i (\theta_1 - \theta_2)}.\]

Step 5: Combine and Simplify the Expression

Combine the simplified magnitude and exponential expressions:\[\frac{z_1}{z_2} = \frac{r_1}{r_2} e^{i (\theta_1 - \theta_2)}.\]

Step 6: Convert Back to Trigonometric Form

Using Euler's formula, convert back to trigonometric form:\[e^{i (\theta_1 - \theta_2)} = \cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2).\]

Step 7: Write the Final Expression for the Quotient

The expression becomes:\[\frac{z_1}{z_2} = \frac{r_1}{r_2} \left[\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2)\right].\]

Step 8: Conclusion

The statement \(\frac{z_1}{z_2} = \frac{r_1}{r_2} \left[\cos \left(\theta_1-\theta_2\right) + i \sin \left(\theta_1-\theta_2\right)\right]\) is shown to be true by the properties of exponentials and Euler's formula.

Key Concepts

Complex Division

Complex division involves dividing one complex number by another. Let's consider two complex numbers in their exponential and trigonometric forms, such as \( z_1 = r_1 e^{i \theta_1} \) and \( z_2 = r_2 e^{i \theta_2} \). By dividing these numbers, we aim to simplify the expression \( \frac{z_1}{z_2} \) into a form that's easy to understand.

Firstly, division of complex numbers in exponential form resembles basic division, but it also involves dividing magnitudes and subtracting angles. A

• Divide the magnitudes to get \( \frac{r_1}{r_2} \).
• Subtract the angles: \( i(\theta_1 - \theta_2) \).

With these, the quotient becomes \( \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)} \). We then use Euler's formula to transform this exponential form back into trigonometric form, showing that complex division can be visualized as manipulating the sizes and angles of vectors in the complex plane.

Euler's Formula

Euler's formula is a key tool when working with complex numbers in exponential form. It beautifully bridges the gap between trigonometric functions and exponential functions. The formula states: \( e^{i \theta} = \cos \theta + i \sin \theta \).

This expression shows how exponential relationships closely correlate with rotations in a complex plane. Particularly, when you multiply by \( e^{i \theta} \), it effectively rotates a vector by angle \( \theta \) in the complex plane. It’s a clean and powerful way to express sine and cosine functions, often simplifying multiplication and division of complex numbers.
Often, Euler's formula helps convert between complex trigonometric expressions and exponential forms, making calculations more straightforward. Using it, the trigonometric form \( \cos(\theta) + i\sin(\theta) \) and the exponential form \( e^{i\theta} \) become interchangeable, offering different perspectives on the same complex numbers.

Trigonometric Form

Trigonometric form represents a complex number using angles and magnitudes, expressed as \( r(\cos \theta + i \sin \theta) \). Here, \( r \) is the magnitude, representing the distance from the origin to the point in the complex plane, and \( \theta \) is the angle which the line to that point makes with the positive x-axis.

This form is particularly useful when visualizing or graphing complex numbers since it lends itself to easily interpreting the geometric representation. Perfectly suited for multiplication and division, the magnitudes multiply or divide, and the angles add or subtract. Therefore, expressing complex numbers in trigonometric form helps clear up and simplify calculations.

• Visualizing complex numbers: identifies magnitude and exact angle.
• Simple multiplication/division: works through adding/subtracting corresponding angles.
• Flexibility: serves well alongside exponential forms using Euler's formula.

Understanding this interplay between vector magnitude and angle is crucial for mastering complex number manipulations.

Exponential Form

Exponential form of a complex number simplifies expressions and calculations, leveraging Euler's formula to condense complex numbers into a neat product of a magnitude and an exponential. Written as \( z = r e^{i\theta} \), where \( r \) and \( \theta \) are as defined in the trigonometric form.

This form elegantly encapsulates the complex number's essence, representing both magnitude and angle in a streamlined manner.

• Makes complex calculations efficient and compact, especially multiplication and division.
• An intuitive approach to problems involving periodic phenomena due to its continuous nature.
• Often used in electrical engineering and physics for dealing with signals and waves.

When solving complex numbers, exponential form often emerges as a preferred method for its straightforward application of the fundamental properties of exponents and Euler's formula. It effectively simplifies otherwise cumbersome multiplication or division.