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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Question
![To convert the given point from cylindrical coordinates to rectangular coordinates, we start with the cylindrical coordinates \((r, \theta, z)\) given as \((6, -\frac{3\pi}{2}, 3)\).
The formula to convert cylindrical coordinates to rectangular coordinates \((x, y, z)\) is:
\[
x = r \cdot \cos(\theta)
\]
\[
y = r \cdot \sin(\theta)
\]
\[
z = z
\]
Given the cylindrical coordinates:
- \(r = 6\)
- \(\theta = -\frac{3\pi}{2}\)
- \(z = 3\)
Calculate \(x\) and \(y\):
\[
x = 6 \cdot \cos\left(-\frac{3\pi}{2}\right)
\]
\[
y = 6 \cdot \sin\left(-\frac{3\pi}{2}\right)
\]
\[
z = 3
\]
Calculate:
- \(\cos\left(-\frac{3\pi}{2}\right)\) = 0
- \(\sin\left(-\frac{3\pi}{2}\right)\) = 1
Substitute these values:
\[
x = 6 \cdot 0 = 0
\]
\[
y = 6 \cdot 1 = 6
\]
Thus, the rectangular coordinates are \((x, y, z) = (0, 6, 3)\).
The diagram depicts a conversion process from cylindrical coordinates to rectangular coordinates, illustrating the transformation step, displaying the empty space for calculation, and marking it with a cross to indicate the need for transformation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fed7d3a8d-0307-419a-b7ca-ff8ecf6a396f%2Fb03cb22a-d54c-48d9-b6b8-a170b2a4fc70%2F9aam14_processed.png&w=3840&q=75)
Transcribed Image Text:To convert the given point from cylindrical coordinates to rectangular coordinates, we start with the cylindrical coordinates \((r, \theta, z)\) given as \((6, -\frac{3\pi}{2}, 3)\).
The formula to convert cylindrical coordinates to rectangular coordinates \((x, y, z)\) is:
\[
x = r \cdot \cos(\theta)
\]
\[
y = r \cdot \sin(\theta)
\]
\[
z = z
\]
Given the cylindrical coordinates:
- \(r = 6\)
- \(\theta = -\frac{3\pi}{2}\)
- \(z = 3\)
Calculate \(x\) and \(y\):
\[
x = 6 \cdot \cos\left(-\frac{3\pi}{2}\right)
\]
\[
y = 6 \cdot \sin\left(-\frac{3\pi}{2}\right)
\]
\[
z = 3
\]
Calculate:
- \(\cos\left(-\frac{3\pi}{2}\right)\) = 0
- \(\sin\left(-\frac{3\pi}{2}\right)\) = 1
Substitute these values:
\[
x = 6 \cdot 0 = 0
\]
\[
y = 6 \cdot 1 = 6
\]
Thus, the rectangular coordinates are \((x, y, z) = (0, 6, 3)\).
The diagram depicts a conversion process from cylindrical coordinates to rectangular coordinates, illustrating the transformation step, displaying the empty space for calculation, and marking it with a cross to indicate the need for transformation.
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