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What are of the point Coordinates the rectangular coordinates whose cynd cylindrical are (r = 2,₁ 0² = ²/= x₁ 2 =7)? ñ X = t Y = T 2=1

What are of the point Coordinates the rectangular coordinates whose cynd cylindrical are (r = 2,₁ 0² = ²/= x₁ 2 =7)? ñ X = t Y = T 2=1

Calculus: Early Transcendentals
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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**Topic: Conversion of Cylindrical to Rectangular Coordinates** ### Problem Statement: What are the rectangular coordinates of the point whose cylindrical coordinates are \((r = 2, \theta = \frac{2\pi}{3}, z = 7)\)? **Rectangular Coordinates Formula:** To convert from cylindrical to rectangular coordinates, use the following formulas: - \( x = r \cdot \cos(\theta) \) - \( y = r \cdot \sin(\theta) \) - \( z = z \) **Calculation Steps:** 1. **Given:** - \( r = 2 \) - \( \theta = \frac{2\pi}{3} \) - \( z = 7 \) 2. **Convert to Rectangular Coordinates:** - Calculate \( x \): \[ x = 2 \cdot \cos\left(\frac{2\pi}{3}\right) \] - Calculate \( y \): \[ y = 2 \cdot \sin\left(\frac{2\pi}{3}\right) \] - \( z \) remains the same: \[ z = 7 \] **Solution Box:** - \( x = \) [calculate result] - \( y = \) [calculate result] - \( z = 7 \) **Note:** Calculate the trigonometric values for \(\cos\left(\frac{2\pi}{3}\right)\) and \(\sin\left(\frac{2\pi}{3}\right)\) to find the exact rectangular coordinates.
Transcribed Image Text:**Topic: Conversion of Cylindrical to Rectangular Coordinates** ### Problem Statement: What are the rectangular coordinates of the point whose cylindrical coordinates are \((r = 2, \theta = \frac{2\pi}{3}, z = 7)\)? **Rectangular Coordinates Formula:** To convert from cylindrical to rectangular coordinates, use the following formulas: - \( x = r \cdot \cos(\theta) \) - \( y = r \cdot \sin(\theta) \) - \( z = z \) **Calculation Steps:** 1. **Given:** - \( r = 2 \) - \( \theta = \frac{2\pi}{3} \) - \( z = 7 \) 2. **Convert to Rectangular Coordinates:** - Calculate \( x \): \[ x = 2 \cdot \cos\left(\frac{2\pi}{3}\right) \] - Calculate \( y \): \[ y = 2 \cdot \sin\left(\frac{2\pi}{3}\right) \] - \( z \) remains the same: \[ z = 7 \] **Solution Box:** - \( x = \) [calculate result] - \( y = \) [calculate result] - \( z = 7 \) **Note:** Calculate the trigonometric values for \(\cos\left(\frac{2\pi}{3}\right)\) and \(\sin\left(\frac{2\pi}{3}\right)\) to find the exact rectangular coordinates.
**Conversion from Cylindrical to Rectangular Coordinates** **Problem Statement:** Determine the rectangular coordinates of a point, given its cylindrical coordinates: - \( r = 2 \) - \( \theta = \frac{\pi}{2} \) - \( z = 7 \) **Solution:** The formulas to convert cylindrical coordinates \((r, \theta, z)\) to rectangular coordinates \((x, y, z)\) are as follows: - \( x = r \cdot \cos(\theta) \) - \( y = r \cdot \sin(\theta) \) - \( z = z \) **Given:** - \( r = 2 \) - \( \theta = \frac{\pi}{2} \) (90 degrees) - \( z = 7 \) **Calculations:** 1. **Calculate \( x \):** \[ x = 2 \cdot \cos\left(\frac{\pi}{2}\right) = 2 \cdot 0 = 0 \] 2. **Calculate \( y \):** \[ y = 2 \cdot \sin\left(\frac{\pi}{2}\right) = 2 \cdot 1 = 2 \] 3. **The \( z \) coordinate remains unchanged:** \[ z = 7 \] **Result:** The rectangular coordinates are: - \( x = 0 \) - \( y = 2 \) - \( z = 7 \) **Rectangular Coordinates:** - \( X = \) [ ] - \( Y = \) [ ] - \( Z = \) [ ] Fill in the boxes with the calculated values: \( X = 0 \), \( Y = 2 \), \( Z = 7 \).
Transcribed Image Text:**Conversion from Cylindrical to Rectangular Coordinates** **Problem Statement:** Determine the rectangular coordinates of a point, given its cylindrical coordinates: - \( r = 2 \) - \( \theta = \frac{\pi}{2} \) - \( z = 7 \) **Solution:** The formulas to convert cylindrical coordinates \((r, \theta, z)\) to rectangular coordinates \((x, y, z)\) are as follows: - \( x = r \cdot \cos(\theta) \) - \( y = r \cdot \sin(\theta) \) - \( z = z \) **Given:** - \( r = 2 \) - \( \theta = \frac{\pi}{2} \) (90 degrees) - \( z = 7 \) **Calculations:** 1. **Calculate \( x \):** \[ x = 2 \cdot \cos\left(\frac{\pi}{2}\right) = 2 \cdot 0 = 0 \] 2. **Calculate \( y \):** \[ y = 2 \cdot \sin\left(\frac{\pi}{2}\right) = 2 \cdot 1 = 2 \] 3. **The \( z \) coordinate remains unchanged:** \[ z = 7 \] **Result:** The rectangular coordinates are: - \( x = 0 \) - \( y = 2 \) - \( z = 7 \) **Rectangular Coordinates:** - \( X = \) [ ] - \( Y = \) [ ] - \( Z = \) [ ] Fill in the boxes with the calculated values: \( X = 0 \), \( Y = 2 \), \( Z = 7 \).
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