Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Complex Number Problem**
**Problem Statement:**
Given \( z = 4 (\cos 35^\circ + i \sin 35^\circ) \), find \( z^6 \). Write your answer in **rectangular form**.
**Explanation:**
This problem involves calculating the sixth power of a complex number given in polar form, and then converting it to rectangular form. The polar form is expressed as \( r (\cos \theta + i \sin \theta) \), where \( r \) is the magnitude and \( \theta \) is the angle.
You will use De Moivre’s Theorem to calculate \( z^6 \):
\[
z^6 = [r (\cos \theta + i \sin \theta)]^6 = r^6 \left( \cos(6\theta) + i \sin(6\theta) \right)
\]
Then, convert this result into rectangular form by calculating the cosine and sine components.
**Note:** Ensure you convert angles properly and evaluate trigonometric functions to get the final rectangular form.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4167b07b-ab2c-42e3-ab89-9f12261de975%2F3e0f025e-c1ef-42df-b101-8819ddc93c7c%2Fudijlbb_processed.png&w=3840&q=75)
Transcribed Image Text:**Complex Number Problem**
**Problem Statement:**
Given \( z = 4 (\cos 35^\circ + i \sin 35^\circ) \), find \( z^6 \). Write your answer in **rectangular form**.
**Explanation:**
This problem involves calculating the sixth power of a complex number given in polar form, and then converting it to rectangular form. The polar form is expressed as \( r (\cos \theta + i \sin \theta) \), where \( r \) is the magnitude and \( \theta \) is the angle.
You will use De Moivre’s Theorem to calculate \( z^6 \):
\[
z^6 = [r (\cos \theta + i \sin \theta)]^6 = r^6 \left( \cos(6\theta) + i \sin(6\theta) \right)
\]
Then, convert this result into rectangular form by calculating the cosine and sine components.
**Note:** Ensure you convert angles properly and evaluate trigonometric functions to get the final rectangular form.
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