Find Each Quotient And Express It In Rectangular Form.1. 2\left(\cos 150^{\circ}+i \sin 150^{\circ}\right) \div 8\left(\cos 210^{\circ}+i \sin 210^{\circ}\right ]Possible Answers:A. − 0.22 + 0.12 I -0.22+0.12i − 0.22 + 0.12 I B. $\frac{1}{4}\left[\sin

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Introduction

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De Moivre's Theorem is a fundamental concept in mathematics that deals with the representation of complex numbers in polar form. It states that for any complex number written in polar form as $r(\cos \theta + i \sin \theta)$, the $n$th power of this number can be expressed as $r^n(\cos n\theta + i \sin n\theta)$. In this article, we will explore the application of De Moivre's Theorem to find the quotient of two complex numbers written in polar form.

Polar Form of Complex Numbers

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A complex number $z$ can be represented in polar form as $z = r(\cos \theta + i \sin \theta)$, where $r$ is the magnitude or modulus of the complex number, and $\theta$ is the argument or angle of the complex number. The polar form of a complex number is a powerful tool for simplifying complex calculations involving complex numbers.

De Moivre's Theorem

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De Moivre's Theorem states that for any complex number written in polar form as $r(\cos \theta + i \sin \theta)$, the $n$th power of this number can be expressed as $r^n(\cos n\theta + i \sin n\theta)$. This theorem is a fundamental concept in mathematics and has numerous applications in various fields, including trigonometry, algebra, and calculus.

Quotient of Complex Numbers

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The quotient of two complex numbers written in polar form can be found using De Moivre's Theorem. Given two complex numbers $z_1 = r_1(\cos \theta_1 + i \sin \theta_1)$ and $z_2 = r_2(\cos \theta_2 + i \sin \theta_2)$, the quotient of these two numbers can be expressed as:

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

Using De Moivre's Theorem, we can rewrite the quotient as:

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

Example Problem

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Find each quotient and express it in rectangular form:

$$2\left(\cos 150^{\circ}+i \sin 150^{\circ}\right) \div 8\left(\cos 210^{\circ}+i \sin 210^{\circ}\right)$$

Solution

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To find the quotient, we can use the formula for the quotient of two complex numbers in polar form:

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

In this case, we have:

$$r_1 = 2, \quad \theta_1 = 150^{\circ}, \quad r_2 = 8, \quad \theta_2 = 210^{\circ}$$

Substituting these values into the formula, we get:

$$\frac{2\left(\cos 150^{\circ}+i \sin 150^{\circ}\right)}{8\left(\cos 210^{\circ}+i \sin 210^{\circ}\right)} = \frac{2}{8} \left(\cos (150^{\circ} - 210^{\circ}) + i \sin (150^{\circ} - 210^{\circ})\right)$$

Simplifying the expression, we get:

$$\frac{2\left(\cos 150^{\circ}+i \sin 150^{\circ}\right)}{8\left(\cos 210^{\circ}+i \sin 210^{\circ}\right)} = \frac{1}{4} \left(\cos (-60^{\circ}) + i \sin (-60^{\circ})\right)$$

Using the properties of cosine and sine functions, we can rewrite the expression as:

$$\frac{2\left(\cos 150^{\circ}+i \sin 150^{\circ}\right)}{8\left(\cos 210^{\circ}+i \sin 210^{\circ}\right)} = \frac{1}{4} \left(\cos 60^{\circ} - i \sin 60^{\circ}\right)$$

Evaluating the cosine and sine functions, we get:

$$\frac{2\left(\cos 150^{\circ}+i \sin 150^{\circ}\right)}{8\left(\cos 210^{\circ}+i \sin 210^{\circ}\right)} = \frac{1}{4} \left(\frac{1}{2} - i \frac{\sqrt{3}}{2}\right)$$

Simplifying the expression, we get:

$$\frac{2\left(\cos 150^{\circ}+i \sin 150^{\circ}\right)}{8\left(\cos 210^{\circ}+i \sin 210^{\circ}\right)} = \frac{1}{8} - \frac{\sqrt{3}}{8}i$$

Conclusion

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In this article, we have explored the application of De Moivre's Theorem to find the quotient of two complex numbers written in polar form. We have used the formula for the quotient of two complex numbers in polar form to find the quotient of two complex numbers, and we have simplified the expression to obtain the final answer. The final answer is $\boxed{\frac{1}{8} - \frac{\sqrt{3}}{8}i}$.

Possible Answers

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A. $-0.22+0.12i$ B. $\frac{1}{4}\left[\sin 30^{\circ}+i \cos 30^{\circ}\right]$ C. $\frac{1}{4}\left[\cos 30^{\circ}-i \sin 30^{\circ}\right]$ D. $\frac{1}{8} - \frac{\sqrt{3}}{8}i$

The correct answer is D. $\frac{1}{8} - \frac{\sqrt{3}}{8}i$.

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Introduction

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In our previous article, we explored the application of De Moivre's Theorem to find the quotient of two complex numbers written in polar form. We used the formula for the quotient of two complex numbers in polar form to find the quotient of two complex numbers, and we simplified the expression to obtain the final answer. In this article, we will answer some frequently asked questions about De Moivre's Theorem and the quotient of complex numbers.

Q&A

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Q: What is De Moivre's Theorem?

A: De Moivre's Theorem is a fundamental concept in mathematics that deals with the representation of complex numbers in polar form. It states that for any complex number written in polar form as $r(\cos \theta + i \sin \theta)$, the $n$th power of this number can be expressed as $r^n(\cos n\theta + i \sin n\theta)$.

Q: How do I apply De Moivre's Theorem to find the quotient of two complex numbers?

A: To apply De Moivre's Theorem to find the quotient of two complex numbers, you can use the formula for the quotient of two complex numbers in polar form:

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

Q: What is the difference between the quotient of two complex numbers and the product of two complex numbers?

A: The quotient of two complex numbers is the result of dividing one complex number by another, while the product of two complex numbers is the result of multiplying two complex numbers together. The formula for the quotient of two complex numbers in polar form is different from the formula for the product of two complex numbers in polar form.

Q: Can I use De Moivre's Theorem to find the square root of a complex number?

A: Yes, you can use De Moivre's Theorem to find the square root of a complex number. To do this, you can use the formula for the square root of a complex number in polar form:

$$\sqrt{z} = \sqrt{r} \left(\cos \frac{\theta}{2} + i \sin \frac{\theta}{2}\right)$$

Q: What are some common mistakes to avoid when applying De Moivre's Theorem?

A: Some common mistakes to avoid when applying De Moivre's Theorem include:

• Not using the correct formula for the quotient of two complex numbers in polar form
• Not simplifying the expression correctly
• Not using the correct values for the magnitude and argument of the complex numbers
• Not checking the final answer for errors

Conclusion

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In this article, we have answered some frequently asked questions about De Moivre's Theorem and the quotient of complex numbers. We have provided explanations and examples to help you understand the concepts and apply them correctly. By following the tips and avoiding common mistakes, you can use De Moivre's Theorem to find the quotient of two complex numbers and solve problems involving complex numbers.

Additional Resources

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• De Moivre's Theorem: A Comprehensive Guide
• Complex Numbers: A Beginner's Guide
• Polar Form of Complex Numbers: A Tutorial

Final Thoughts

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De Moivre's Theorem is a powerful tool for working with complex numbers. By understanding the concepts and applying them correctly, you can solve problems involving complex numbers and find the quotient of two complex numbers. Remember to use the correct formula, simplify the expression correctly, and check the final answer for errors. With practice and patience, you can become proficient in using De Moivre's Theorem to solve complex problems.