Problem 19 - Precalculus

Question

Find the quotient \(\frac{z_{1}}{z_{2}}\) and express it in rectangular form. $$z_{1}=45\left[\cos \left(\frac{22 \pi}{15}\right)+i \sin \left(\frac{22 \pi}{15}\right)\right] \text { and } z_{2}=9\left[\cos \left(\frac{2 \pi}{15}\right)+i \sin \left(\frac{2 \pi}{15}\right)\right]$$

Step-by-Step Solution

Answer

The quotient is \(-\frac{5}{2} - \frac{5\sqrt{3}}{2}i\).

Step 1: Convert to Exponential Form

Convert the given complex numbers to exponential form using Euler's formula. Define the complex numbers as follows:\[ z_1 = 45 \left[ \cos \left(\frac{22 \pi}{15}\right) + i \sin \left(\frac{22 \pi}{15}\right) \right] = 45 e^{i \frac{22 \pi}{15}} \]and \[ z_2 = 9 \left[ \cos \left(\frac{2 \pi}{15}\right) + i \sin \left(\frac{2 \pi}{15}\right) \right] = 9 e^{i \frac{2 \pi}{15}} \]

Step 2: Apply Division Formula for Complex Numbers

To find the quotient \( \frac{z_1}{z_2} \), divide the magnitudes and subtract the exponents:\[ \frac{z_1}{z_2} = \frac{45}{9} e^{i \left( \frac{22\pi}{15} - \frac{2\pi}{15} \right)} \]Simplify the expression:\[ \frac{z_1}{z_2} = 5 e^{i \frac{20 \pi}{15}} = 5 e^{i \frac{4 \pi}{3}} \]

Step 3: Convert Back to Rectangular Form

Use Euler's formula again to convert \( 5 e^{i \frac{4\pi}{3}} \) to rectangular form. First, find the cosine and sine:\[ \cos\left(\frac{4\pi}{3}\right) = -\frac{1}{2}, \quad \sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2} \]Then, substitute back:\[ z = 5 \left[ -\frac{1}{2} + i \left(-\frac{\sqrt{3}}{2}\right) \right] \]Simplify:\[ z = -\frac{5}{2} - \frac{5\sqrt{3}}{2} i \]

Step 4: Conclusion

Thus, the quotient \( \frac{z_1}{z_2} \) in rectangular form is:\[ -\frac{5}{2} - \frac{5\sqrt{3}}{2}i \]

Key Concepts

Complex Number Division

Complex number division sounds a bit intimidating at first, but it's similar to dividing regular numbers. Only, we deal with both magnitude and direction more consciously. In our original exercise, we are tasked with finding the quotient of two complex numbers, \( z_1 \) and \( z_2 \). These numbers are expressed in what we call the polar form: a form that emphasizes their magnitude and angular component.

Here's the basic strategy we apply when dividing complex numbers in polar form:

• Divide the magnitudes: The magnitudes of the complex numbers are the coefficients. So, for our example, \( \frac{45}{9} = 5 \).
• Subtract the angles: The angles in the exponential terms need to be subtracted. Hence, we compute \( \frac{22\pi}{15} - \frac{2\pi}{15} = \frac{20\pi}{15} = \frac{4\pi}{3} \).

This gives us a new complex number in exponential form (\( 5 e^{i\frac{4\pi}{3}} \)). Understanding this process makes dealing with complex numbers much simpler!

Polar to Rectangular Conversion

After working with the division of complex numbers in their polar form, the next task involves converting back to rectangular form, which is often easier to visualize. The rectangular form is the standard form of a complex number, expressed as \( a + bi \) where \( a \) and \( b \) are real numbers.

To convert from polar (or exponential) form to rectangular form, we make use of Euler's formula. The critical expressions here are \( \cos(\theta) \) and \( \sin(\theta) \) which correspond to the real and imaginary parts, respectively:

• \( 5 \cos\left(\frac{4\pi}{3}\right) = 5 \left(-\frac{1}{2}\right) = -\frac{5}{2} \)
• \( 5 \sin\left(\frac{4\pi}{3}\right) = 5 \left(-\frac{\sqrt{3}}{2}\right) = -\frac{5\sqrt{3}}{2} \)

So the complex number in rectangular form is \( -\frac{5}{2} - \frac{5\sqrt{3}}{2}i \). This transformation is essential because it shifts our perspective from abstract angles back to tangible coordinates on the complex plane.

Euler's Formula

Euler's formula is a wonderful bridge between the trigonometric and exponential form of complex numbers. It's expressed as \( e^{i\theta} = \cos(\theta) + i\sin(\theta) \).

In the exercise solution, we used Euler's formula to switch between forms of complex numbers seamlessly:

• While in polar/exponential form, it simplifies calculations such as multiplication and division.
• During the conversion back to rectangular form, it becomes straightforward to identify the cosine and sine components -- perfect for easy calculation of the real and imaginary parts.

Euler's formula not only simplifies mathematical operations but also encapsulates a deep, elegant connection between two fundamental mathematical concepts — displacement and rotation — in the complex number plane. Understanding this formula can enhance your handling of complex numbers considerably.