Finding Z^5 In Rectangular Form A Step By Step Guide

Let's delve into the process of finding $z^5$ in rectangular form, given that z = 2 ext{ cis } rac{\pi}{6}. This problem combines concepts from complex numbers, De Moivre's Theorem, and trigonometric identities. Our goal is to express $z^5$ in the form $a + bi$, where $a$ and $b$ are real numbers. We'll break this down step by step to ensure clarity and understanding.

Understanding the Problem

First, we need to understand what z = 2 \text{ cis } rac{\pi}{6} means. The notation "cis" stands for "cosine plus i sine," so we can rewrite $z$ as:

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

Here, $2$ is the modulus (or magnitude) of the complex number, and $\frac{\pi}{6}$ is the argument (or angle) in radians. To find $z^5$, we will use De Moivre's Theorem, which is a powerful tool for raising complex numbers in polar form to a power. After applying De Moivre's Theorem, we will convert the result back into rectangular form.

Applying De Moivre's Theorem

De Moivre's Theorem states that for any complex number in polar form $r(\cos \theta + i \sin \theta)$ and any integer $n$:

$[r(\cos \theta + i \sin \theta)]^n = r^n(\cos n\theta + i \sin n\theta)$

In our case, $r = 2$, $\theta = \frac{\pi}{6}$, and $n = 5$. Applying De Moivre's Theorem, we get:

$z^5 = \left[ 2 \left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right) \right]^5$ $z^5 = 2^5 \left( \cos \left( 5 \cdot \frac{\pi}{6} \right) + i \sin \left( 5 \cdot \frac{\pi}{6} \right) \right)$ $z^5 = 32 \left( \cos \frac{5\pi}{6} + i \sin \frac{5\pi}{6} \right)$

Now, we need to evaluate the cosine and sine of $\frac{5\pi}{6}$.

Evaluating Trigonometric Functions

The angle $\frac{5\pi}{6}$ is in the second quadrant, where cosine is negative and sine is positive. We can relate $\frac{5\pi}{6}$ to the reference angle in the first quadrant, which is $\pi - \frac{5\pi}{6} = \frac{\pi}{6}$. We know the values of cosine and sine for $\frac{\pi}{6}$:

$\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$ $\sin \frac{\pi}{6} = \frac{1}{2}$

Therefore:

$\cos \frac{5\pi}{6} = -\cos \frac{\pi}{6} = -\frac{\sqrt{3}}{2}$ $\sin \frac{5\pi}{6} = \sin \frac{\pi}{6} = \frac{1}{2}$

Substituting these values back into the expression for $z^5$, we have:

$z^5 = 32 \left( -\frac{\sqrt{3}}{2} + i \frac{1}{2} \right)$

Converting to Rectangular Form

Now, we distribute the $32$ to get the rectangular form:

$z^5 = 32 \left( -\frac{\sqrt{3}}{2} \right) + 32 \left( i \frac{1}{2} \right)$ $z^5 = -16\sqrt{3} + 16i$

Thus, $z^5$ in rectangular form is $-16\sqrt{3} + 16i$.

Conclusion

By applying De Moivre's Theorem and evaluating the trigonometric functions, we found that $z^5 = -16\sqrt{3} + 16i$. This matches option E. Understanding complex numbers and their properties, along with trigonometric identities, is crucial for solving such problems. The ability to convert between polar and rectangular forms is a fundamental skill in complex number arithmetic.

Detailed Solution and Explanation

Step 1: Convert to Polar Form

The given complex number is z = 2 ext{ cis } rac{\pi}{6}. This is already in polar form, where the modulus (or magnitude) is $r = 2$ and the argument (or angle) is \theta = rac{\pi}{6}. The notation $\text{cis } \theta$ is shorthand for $\cos \theta + i \sin \theta$, so we can write $z$ as:

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

This form is convenient because it allows us to use De Moivre's Theorem easily.

Step 2: Apply De Moivre's Theorem

De Moivre's Theorem states that for any complex number $z = r(\cos \theta + i \sin \theta)$ and any integer $n$, we have:

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

In this case, we want to find $z^5$, so we have $n = 5$. Applying the theorem:

$z^5 = \left[ 2 \left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right) \right]^5$ $z^5 = 2^5 \left( \cos \left( 5 \cdot \frac{\pi}{6} \right) + i \sin \left( 5 \cdot \frac{\pi}{6} \right) \right)$ $z^5 = 32 \left( \cos \frac{5\pi}{6} + i \sin \frac{5\pi}{6} \right)$

Step 3: Evaluate Trigonometric Functions

We need to find the values of $\cos \frac{5\pi}{6}$ and $\sin \frac{5\pi}{6}$. The angle $\frac{5\pi}{6}$ is in the second quadrant. In the second quadrant, cosine is negative, and sine is positive. The reference angle for $\frac{5\pi}{6}$ is the angle it makes with the x-axis, which is $\pi - \frac{5\pi}{6} = \frac{\pi}{6}$.

We know the values of cosine and sine for $\frac{\pi}{6}$:

$\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$ $\sin \frac{\pi}{6} = \frac{1}{2}$

Since $\frac{5\pi}{6}$ is in the second quadrant:

$\cos \frac{5\pi}{6} = -\cos \frac{\pi}{6} = -\frac{\sqrt{3}}{2}$ $\sin \frac{5\pi}{6} = \sin \frac{\pi}{6} = \frac{1}{2}$

Step 4: Convert to Rectangular Form

Substitute the values of the trigonometric functions back into the expression for $z^5$:

$z^5 = 32 \left( -\frac{\sqrt{3}}{2} + i \frac{1}{2} \right)$

Now, distribute the $32$:

$z^5 = 32 \cdot \left( -\frac{\sqrt{3}}{2} \right) + 32 \cdot \left( i \frac{1}{2} \right)$ $z^5 = -16\sqrt{3} + 16i$

So, $z^5$ in rectangular form is $-16\sqrt{3} + 16i$.

Step 5: Final Answer

Comparing our result with the given options, we see that the correct answer is:

E. $-16\sqrt{3} + 16i$

This step-by-step solution provides a detailed explanation of how to use De Moivre's Theorem and trigonometric identities to find the rectangular form of a complex number raised to a power. Understanding these concepts is essential for mastering complex number arithmetic.

In conclusion, by meticulously applying De Moivre's Theorem and understanding the properties of complex numbers, we've successfully determined that $z^5 = -16\sqrt{3} + 16i$ when $z = 2 \text{ cis } \frac{\pi}{6}$. This exercise underscores the importance of several key concepts in complex number theory and trigonometry.

First and foremost, De Moivre's Theorem provides a powerful and efficient method for raising complex numbers in polar form to integer powers. The theorem elegantly links the modulus and argument of a complex number to its powers, simplifying what could otherwise be a cumbersome multiplication process. Understanding and skillfully applying this theorem is crucial for solving problems involving powers of complex numbers.

Secondly, this problem highlights the significance of converting between polar and rectangular forms of complex numbers. The polar form, represented as $r(\cos \theta + i \sin \theta)$, is particularly useful for multiplication and exponentiation due to De Moivre's Theorem. However, the rectangular form, $a + bi$, is often more convenient for addition and subtraction. The ability to seamlessly transition between these forms is a cornerstone of complex number manipulation. Mastery of these conversions allows for flexibility in problem-solving and provides a deeper insight into the nature of complex numbers.

Additionally, the problem reinforces the importance of trigonometric functions and their values at key angles. The evaluation of $\cos \frac{5\pi}{6}$ and $\sin \frac{5\pi}{6}$ required a solid understanding of the unit circle and the properties of trigonometric functions in different quadrants. Familiarity with these trigonometric values and their relationships is essential for many mathematical contexts, extending beyond complex numbers.

Moreover, this exercise exemplifies the interconnectedness of different mathematical domains. It elegantly combines concepts from complex numbers, trigonometry, and algebra. Successful problem-solving often requires integrating knowledge from various areas, showcasing the holistic nature of mathematics. This integration not only deepens understanding but also enhances the ability to approach novel problems from multiple perspectives.

The implications of complex number theory extend far beyond academic exercises. Complex numbers are fundamental to many areas of science and engineering, including electrical engineering, quantum mechanics, and fluid dynamics. Their ability to represent oscillations and wave phenomena makes them indispensable in signal processing and Fourier analysis. Therefore, a thorough understanding of complex numbers is not just an academic pursuit but a vital skill for many professionals.

In summary, the process of finding $z^5$ in rectangular form serves as a valuable learning experience. It reinforces the understanding of De Moivre's Theorem, polar and rectangular forms, trigonometric functions, and the interconnectedness of mathematical concepts. Ultimately, mastering these principles not only enhances mathematical proficiency but also unlocks the door to numerous applications in science and engineering. Practice and application of these concepts are key to developing a deep and lasting understanding.

To solidify your understanding of complex numbers and De Moivre's Theorem, try working through these practice problems. These problems cover similar concepts and techniques as the example we worked through, providing an opportunity to apply what you've learned. Consistent practice is crucial for mastering mathematical skills, and these problems will help you build confidence and proficiency.

1. Problem 1: For $z = 3 \text{ cis } \frac{\pi}{4}$, find $z^3$ in rectangular form.

   • This problem is similar to the example, requiring you to apply De Moivre's Theorem and convert the result back to rectangular form. Pay close attention to the trigonometric values of the angles involved.

2. Problem 2: If $z = 1 + i$, find $z^4$ in rectangular form.

   • In this problem, you'll first need to convert the complex number from rectangular form to polar form. Then, apply De Moivre's Theorem and convert back to rectangular form. This problem emphasizes the importance of being able to switch between the two forms.

3. Problem 3: Let $z = 2 \text{ cis } \frac{2\pi}{3}$. Determine $z^6$ in rectangular form.

   • This problem involves a larger power, so be careful with the calculations. Remember to simplify the angle after applying De Moivre's Theorem, if necessary, and evaluate the trigonometric functions correctly.

4. Problem 4: Find $z^5$ in rectangular form, given $z = \sqrt{2} \left( \cos \frac{7\pi}{4} + i \sin \frac{7\pi}{4} \right)$.

   • This problem provides the complex number in polar form, but the angle is greater than $2\pi$. You may want to find a coterminal angle (an angle that differs by a multiple of $2\pi$) to simplify the calculations.

5. Problem 5: If $z = -1 - i\sqrt{3}$, find $z^3$ in rectangular form.

   • Similar to Problem 2, you'll need to convert from rectangular to polar form first. Pay attention to the quadrant of the complex number when finding the argument (angle).

Working through these problems will not only help you practice applying De Moivre's Theorem but also reinforce your understanding of complex number arithmetic and trigonometry. Remember to break each problem down into steps, carefully apply the formulas, and double-check your calculations. Good luck, and happy problem-solving!

By consistently practicing and applying these concepts, you will build a robust understanding of complex numbers and their applications.