Write $z_1$ And $z_2$ In Polar Form, And Then Find The Product $ Z 1 Z 2 Z_1 Z_2 Z 1 ​ Z 2 ​ [/tex] And The Quotients $z_1 / Z_2$ And $1 / Z_1$. Express Your Answers In Polar Form.57. $z_1 = \sqrt{3} + I, ,

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Introduction

In this article, we will explore the process of converting complex numbers from rectangular form to polar form and then perform various operations such as multiplication and division. We will start by converting the given complex numbers $z_1 = \sqrt{3} + i$ and $z_2 = 2 - i$ into polar form. Then, we will find the product $z_1 z_2$, the quotients $z_1 / z_2$ and $1 / z_1$, and express our answers in polar form.

Converting Complex Numbers to Polar Form

To convert a complex number from rectangular form to polar form, we use the following formulas:

$$r = \sqrt{x^2 + y^2}$$

$$\theta = \tan^{-1}\left(\frac{y}{x}\right)$$

where $x$ and $y$ are the real and imaginary parts of the complex number, respectively.

Converting $z_1 = \sqrt{3} + i$ to Polar Form

We can calculate the magnitude $r$ and angle $\theta$ of $z_1$ as follows:

$$r = \sqrt{(\sqrt{3})^2 + 1^2} = \sqrt{3 + 1} = \sqrt{4} = 2$$

$$\theta = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \tan^{-1}\left(\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}\right) = \tan^{-1}\left(\frac{\sqrt{3}}{3}\right) = \frac{\pi}{6}$$

Therefore, the polar form of $z_1$ is:

$$z_1 = 2\left(\cos\left(\frac{\pi}{6}\right) + i\sin\left(\frac{\pi}{6}\right)\right)$$

Converting $z_2 = 2 - i$ to Polar Form

We can calculate the magnitude $r$ and angle $\theta$ of $z_2$ as follows:

$$r = \sqrt{2^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5}$$

$$\theta = \tan^{-1}\left(\frac{-1}{2}\right) = \tan^{-1}\left(\frac{-1}{2} \cdot \frac{1}{-1}\right) = \tan^{-1}\left(\frac{1}{2}\right) = \frac{\pi}{6}$$

However, since $z_2$ lies in the fourth quadrant, we need to add $\pi$ to the angle:

$$\theta = \frac{\pi}{6} + \pi = \frac{7\pi}{6}$$

Therefore, the polar form of $z_2$ is:

$$z_2 = \sqrt{5}\left(\cos\left(\frac{7\pi}{6}\right) + i\sin\left(\frac{7\pi}{6}\right)\right)$$

Finding the Product $z_1 z_2$

To find the product of two complex numbers in polar form, we multiply their magnitudes and add their angles:

$$z_1 z_2 = r_1 r_2 \left(\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)\right)$$

Using the polar forms of $z_1$ and $z_2$, we can calculate the product as follows:

$$z_1 z_2 = 2 \cdot \sqrt{5} \left(\cos\left(\frac{\pi}{6} + \frac{7\pi}{6}\right) + i\sin\left(\frac{\pi}{6} + \frac{7\pi}{6}\right)\right)$$

$$z_1 z_2 = 2\sqrt{5} \left(\cos\left(\frac{8\pi}{6}\right) + i\sin\left(\frac{8\pi}{6}\right)\right)$$

$$z_1 z_2 = 2\sqrt{5} \left(\cos\left(\frac{4\pi}{3}\right) + i\sin\left(\frac{4\pi}{3}\right)\right)$$

Finding the Quotients $z_1 / z_2$ and $1 / z_1$

To find the quotient of two complex numbers in polar form, we divide their magnitudes and subtract their angles:

$$\frac{z_1}{z_2} = \frac{r_1}{r_2} \left(\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)\right)$$

Using the polar forms of $z_1$ and $z_2$, we can calculate the quotient as follows:

$$\frac{z_1}{z_2} = \frac{2}{\sqrt{5}} \left(\cos\left(\frac{\pi}{6} - \frac{7\pi}{6}\right) + i\sin\left(\frac{\pi}{6} - \frac{7\pi}{6}\right)\right)$$

$$\frac{z_1}{z_2} = \frac{2}{\sqrt{5}} \left(\cos\left(-\frac{6\pi}{6}\right) + i\sin\left(-\frac{6\pi}{6}\right)\right)$$

$$\frac{z_1}{z_2} = \frac{2}{\sqrt{5}} \left(\cos\left(-\pi\right) + i\sin\left(-\pi\right)\right)$$

$$\frac{z_1}{z_2} = \frac{2}{\sqrt{5}} \left(-1 + i(0)\right)$$

$$\frac{z_1}{z_2} = -\frac{2}{\sqrt{5}}$$

To find the quotient $1 / z_1$, we can use the formula:

$$\frac{1}{z_1} = \frac{1}{r_1} \left(\cos(-\theta_1) + i\sin(-\theta_1)\right)$$

Using the polar form of $z_1$, we can calculate the quotient as follows:

$$\frac{1}{z_1} = \frac{1}{2} \left(\cos\left(-\frac{\pi}{6}\right) + i\sin\left(-\frac{\pi}{6}\right)\right)$$

$$\frac{1}{z_1} = \frac{1}{2} \left(\cos\left(\frac{\pi}{6}\right) - i\sin\left(\frac{\pi}{6}\right)\right)$$

$$\frac{1}{z_1} = \frac{1}{2} \left(\frac{\sqrt{3}}{2} - i\frac{1}{2}\right)$$

$$\frac{1}{z_1} = \frac{\sqrt{3}}{4} - \frac{i}{4}$$

Conclusion

In this article, we have converted the complex numbers $z_1 = \sqrt{3} + i$ and $z_2 = 2 - i$ into polar form. We have then found the product $z_1 z_2$, the quotients $z_1 / z_2$ and $1 / z_1$, and expressed our answers in polar form. The results are:

• $z_1 z_2 = 2\sqrt{5} \left(\cos\left(\frac{4\pi}{3}\right) + i\sin\left(\frac{4\pi}{3}\right)\right)$
• $\frac{z_1}{z_2} = -\frac{2}{\sqrt{5}}$
• $\frac{1}{z_1} = \frac{\sqrt{3}}{4} - \frac{i}{4}$
  

Introduction

In our previous article, we explored the process of converting complex numbers from rectangular form to polar form and then performing various operations such as multiplication and division. In this article, we will answer some frequently asked questions about complex numbers in polar form.

Q: What is the polar form of a complex number?

A: The polar form of a complex number is a way of expressing it in terms of its magnitude (or length) and angle (or direction) from the positive real axis. It is written in the form:

$$z = r(\cos(\theta) + i\sin(\theta))$$

where $r$ is the magnitude and $\theta$ is the angle.

Q: How do I convert a complex number from rectangular form to polar form?

A: To convert a complex number from rectangular form to polar form, you can use the following formulas:

$$r = \sqrt{x^2 + y^2}$$

$$\theta = \tan^{-1}\left(\frac{y}{x}\right)$$

where $x$ and $y$ are the real and imaginary parts of the complex number, respectively.

Q: What is the product of two complex numbers in polar form?

A: The product of two complex numbers in polar form is found by multiplying their magnitudes and adding their angles:

$$z_1 z_2 = r_1 r_2 \left(\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)\right)$$

Q: What is the quotient of two complex numbers in polar form?

A: The quotient of two complex numbers in polar form is found by dividing their magnitudes and subtracting their angles:

$$\frac{z_1}{z_2} = \frac{r_1}{r_2} \left(\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)\right)$$

Q: How do I find the reciprocal of a complex number in polar form?

A: To find the reciprocal of a complex number in polar form, you can use the formula:

$$\frac{1}{z_1} = \frac{1}{r_1} \left(\cos(-\theta_1) + i\sin(-\theta_1)\right)$$

Q: What are some common mistakes to avoid when working with complex numbers in polar form?

A: Some common mistakes to avoid when working with complex numbers in polar form include:

• Not converting the complex number to polar form correctly
• Not multiplying or dividing the magnitudes correctly
• Not adding or subtracting the angles correctly
• Not using the correct formulas for the product, quotient, and reciprocal

Q: How do I apply complex numbers in polar form to real-world problems?

A: Complex numbers in polar form can be applied to a wide range of real-world problems, including:

• Electrical engineering: Complex numbers in polar form are used to represent AC circuits and analyze their behavior.
• Signal processing: Complex numbers in polar form are used to represent signals and analyze their frequency content.
• Control systems: Complex numbers in polar form are used to represent the transfer function of a system and analyze its stability.

Conclusion

In this article, we have answered some frequently asked questions about complex numbers in polar form. We have covered topics such as converting complex numbers to polar form, finding the product and quotient of complex numbers in polar form, and applying complex numbers in polar form to real-world problems. By understanding complex numbers in polar form, you can gain a deeper understanding of complex analysis and apply it to a wide range of fields.

Additional Resources

For further learning, we recommend the following resources:

• "Complex Analysis" by Walter Rudin
• "Complex Variables and Applications" by James W. Brown and Ruel V. Churchill
• "Polar Form of Complex Numbers" by Math Open Reference

We hope this article has been helpful in answering your questions about complex numbers in polar form. If you have any further questions or need additional clarification, please don't hesitate to ask.