Find ( 2 ( Cos ⁡ 2 Π 3 + I Sin ⁡ 2 Π 3 ) ) 5 \left(2\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right)\right)^5 ( 2 ( Cos 3 2 Π ​ + I Sin 3 2 Π ​ ) ) 5 A. − 16 3 − 16 I -16 \sqrt{3}-16 I − 16 3 ​ − 16 I B. − 16 − 16 I 3 -16-16 I \sqrt{3} − 16 − 16 I 3 ​ C. 16 + 16 I 3 16+16 I \sqrt{3} 16 + 16 I 3 ​ D. 16 3 + 16 I 16 \sqrt{3}+16 I 16 3 ​ + 16 I

Introduction

De Moivre's Theorem is a fundamental concept in mathematics that allows us to find the power of a complex number. This theorem is named after the French mathematician Abraham de Moivre, who first discovered it in the 18th century. In this article, we will explore De Moivre's Theorem and use it to find the value of a complex number raised to a power.

What is De Moivre's Theorem?

De Moivre's Theorem states that for any complex number written in polar form as $r(\cos \theta + i \sin \theta)$, the power of this complex number can be found using the following formula:

$$\left(r(\cos \theta + i \sin \theta)\right)^n = r^n(\cos (n\theta) + i \sin (n\theta))$$

This theorem is a powerful tool for finding complex numbers raised to a power, and it has many applications in mathematics and engineering.

Polar Form of Complex Numbers

Before we can apply De Moivre's Theorem, we need to convert the complex number into polar form. The polar form of a complex number is written as $r(\cos \theta + i \sin \theta)$, where $r$ is the magnitude of the complex number and $\theta$ is the angle between the positive x-axis and the line segment connecting the origin to the complex number.

Converting Complex Numbers to Polar Form

To convert a complex number to polar form, we can use the following steps:

1. Find the magnitude of the complex number using the formula $r = \sqrt{a^2 + b^2}$, where $a$ and $b$ are the real and imaginary parts of the complex number.
2. Find the angle between the positive x-axis and the line segment connecting the origin to the complex number using the formula $\theta = \tan^{-1}\left(\frac{b}{a}\right)$.

Applying De Moivre's Theorem

Now that we have the polar form of the complex number, we can apply De Moivre's Theorem to find the power of the complex number. In this case, we are given the complex number $2\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right)$ and we need to find its fifth power.

Using De Moivre's Theorem, we can write:

$$\left(2\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right)\right)^5 = 2^5\left(\cos \left(5\cdot\frac{2 \pi}{3}\right)+i \sin \left(5\cdot\frac{2 \pi}{3}\right)\right)$$

Simplifying the Expression

Now we can simplify the expression using the following steps:

1. Evaluate the magnitude of the complex number: $2^5 = 32$
2. Evaluate the angle: $5\cdot\frac{2 \pi}{3} = \frac{10 \pi}{3}$
3. Evaluate the cosine and sine of the angle: $\cos \frac{10 \pi}{3} = \cos \left(2 \pi - \frac{4 \pi}{3}\right) = \cos \frac{4 \pi}{3} = -\frac{1}{2}$ and $\sin \frac{10 \pi}{3} = \sin \left(2 \pi - \frac{4 \pi}{3}\right) = -\sin \frac{4 \pi}{3} = -\frac{\sqrt{3}}{2}$

Finding the Final Answer

Now we can substitute the values we found into the expression:

$$32\left(-\frac{1}{2}+i \left(-\frac{\sqrt{3}}{2}\right)\right) = -16 - 16i\sqrt{3}$$

Therefore, the final answer is:

-16 - 16i√3

This is the correct answer, and it can be verified by using other methods such as converting the complex number to rectangular form and raising it to the power of 5.

Conclusion

In this article, we used De Moivre's Theorem to find the power of a complex number. We first converted the complex number to polar form, then applied De Moivre's Theorem to find the power of the complex number. Finally, we simplified the expression and found the final answer. This theorem is a powerful tool for finding complex numbers raised to a power, and it has many applications in mathematics and engineering.

References

• De Moivre, A. (1730). "Miscellanea Analytica de Seriebus et Quadraturis." London: G. Woodfall.
• Euler, L. (1748). "Introductio in Analysin Infinitorum." Lausanne: Marc-Michel Bousquet.
• Courant, R. (1937). "Differential and Integral Calculus." New York: Interscience Publishers.

Discussion

This problem is a classic example of how De Moivre's Theorem can be used to find the power of a complex number. The theorem is a powerful tool for solving problems in mathematics and engineering, and it has many applications in fields such as physics, engineering, and computer science.

In this problem, we used De Moivre's Theorem to find the fifth power of a complex number. We first converted the complex number to polar form, then applied De Moivre's Theorem to find the power of the complex number. Finally, we simplified the expression and found the final answer.

If you have any questions or comments about this problem, please feel free to ask. I would be happy to help you understand the solution and provide additional resources for further learning.

Related Problems

• Find the power of the complex number $3\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)$ using De Moivre's Theorem.
• Find the power of the complex number $4\left(\cos \frac{3 \pi}{2}+i \sin \frac{3 \pi}{2}\right)$ using De Moivre's Theorem.
• Find the power of the complex number $5\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)$ using De Moivre's Theorem.

These problems are similar to the one we solved in this article, and they can be solved using the same techniques and formulas. If you have any questions or need help with these problems, please feel free to ask.

Introduction

De Moivre's Theorem is a fundamental concept in mathematics that allows us to find the power of a complex number. In this article, we will answer some common questions about De Moivre's Theorem and provide additional resources for further learning.

Q: What is De Moivre's Theorem?

A: De Moivre's Theorem is a mathematical formula that allows us to find the power of a complex number. It states that for any complex number written in polar form as $r(\cos \theta + i \sin \theta)$, the power of this complex number can be found using the following formula:

$$\left(r(\cos \theta + i \sin \theta)\right)^n = r^n(\cos (n\theta) + i \sin (n\theta))$$

Q: How do I apply De Moivre's Theorem?

A: To apply De Moivre's Theorem, you need to follow these steps:

1. Convert the complex number to polar form.
2. Find the magnitude of the complex number using the formula $r = \sqrt{a^2 + b^2}$, where $a$ and $b$ are the real and imaginary parts of the complex number.
3. Find the angle between the positive x-axis and the line segment connecting the origin to the complex number using the formula $\theta = \tan^{-1}\left(\frac{b}{a}\right)$.
4. Apply De Moivre's Theorem using the formula $\left(r(\cos \theta + i \sin \theta)\right)^n = r^n(\cos (n\theta) + i \sin (n\theta))$.

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 converting the complex number to polar form.
• Not finding the magnitude and angle of the complex number correctly.
• Not applying De Moivre's Theorem correctly using the formula $\left(r(\cos \theta + i \sin \theta)\right)^n = r^n(\cos (n\theta) + i \sin (n\theta))$.

Q: How do I simplify the expression after applying De Moivre's Theorem?

A: To simplify the expression after applying De Moivre's Theorem, you can use the following steps:

1. Evaluate the magnitude of the complex number raised to the power of $n$.
2. Evaluate the cosine and sine of the angle $n\theta$.
3. Substitute the values into the expression and simplify.

Q: What are some real-world applications of De Moivre's Theorem?

A: De Moivre's Theorem has many real-world applications in fields such as physics, engineering, and computer science. Some examples include:

• Finding the power of complex numbers in electrical engineering.
• Solving problems in trigonometry and geometry.
• Modeling population growth and decay in biology.

Q: How do I prove De Moivre's Theorem?

A: De Moivre's Theorem can be proved using the following steps:

1. Start with the formula for the power of a complex number in polar form.
2. Use the binomial theorem to expand the expression.
3. Simplify the expression using trigonometric identities.
4. Show that the resulting expression is equal to the formula for De Moivre's Theorem.

Conclusion

De Moivre's Theorem is a fundamental concept in mathematics that allows us to find the power of a complex number. In this article, we answered some common questions about De Moivre's Theorem and provided additional resources for further learning. We hope this article has been helpful in understanding De Moivre's Theorem and its applications.

References

• De Moivre, A. (1730). "Miscellanea Analytica de Seriebus et Quadraturis." London: G. Woodfall.
• Euler, L. (1748). "Introductio in Analysin Infinitorum." Lausanne: Marc-Michel Bousquet.
• Courant, R. (1937). "Differential and Integral Calculus." New York: Interscience Publishers.

Discussion

This article is a continuation of our previous article on De Moivre's Theorem. We hope this article has been helpful in understanding De Moivre's Theorem and its applications. If you have any questions or need help with any of the topics discussed in this article, please feel free to ask.

Related Problems

• Find the power of the complex number $3\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)$ using De Moivre's Theorem.
• Find the power of the complex number $4\left(\cos \frac{3 \pi}{2}+i \sin \frac{3 \pi}{2}\right)$ using De Moivre's Theorem.
• Find the power of the complex number $5\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)$ using De Moivre's Theorem.

These problems are similar to the one we solved in this article, and they can be solved using the same techniques and formulas. If you have any questions or need help with these problems, please feel free to ask.