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What are the spe spherical coordinat. whoose rectangular point (1₁ 1₁-5)? of the Coordinates p=0 0=1 4= [ d are

What are the spe spherical coordinat. whoose rectangular point (1₁ 1₁-5)? of the Coordinates p=0 0=1 4= [ d are

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Question**: What are the spherical coordinates of the point whose rectangular coordinates are (4, 1, -5)? **Spherical Coordinates Calculation**: 1. **ρ (rho) =** [Box for answer] 2. **θ (theta) =** [Box for answer] 3. **Φ (phi) =** [Box for answer] To find the spherical coordinates from the rectangular coordinates (x, y, z), use the following formulas: - \( \rho = \sqrt{x^2 + y^2 + z^2} \) - \( \theta = \arctan\left(\frac{y}{x}\right) \) - \( \Phi = \arccos\left(\frac{z}{\rho}\right) \) Apply these formulas to the given coordinates (4, 1, -5). (Note: The given text and setup suggest a problem-solving context for students to practice converting rectangular coordinates to spherical coordinates.)
Transcribed Image Text:**Question**: What are the spherical coordinates of the point whose rectangular coordinates are (4, 1, -5)? **Spherical Coordinates Calculation**: 1. **ρ (rho) =** [Box for answer] 2. **θ (theta) =** [Box for answer] 3. **Φ (phi) =** [Box for answer] To find the spherical coordinates from the rectangular coordinates (x, y, z), use the following formulas: - \( \rho = \sqrt{x^2 + y^2 + z^2} \) - \( \theta = \arctan\left(\frac{y}{x}\right) \) - \( \Phi = \arccos\left(\frac{z}{\rho}\right) \) Apply these formulas to the given coordinates (4, 1, -5). (Note: The given text and setup suggest a problem-solving context for students to practice converting rectangular coordinates to spherical coordinates.)
**Question:** What are the spherical coordinates of the point whose rectangular coordinates are \((4, 1, -5)\)? **Solution:** To find the spherical coordinates \((\rho, \theta, \phi)\), we use the following conversions from rectangular coordinates \((x, y, z)\): 1. \(\rho = \sqrt{x^2 + y^2 + z^2}\) 2. \(\theta = \tan^{-1} \left(\frac{y}{x}\right)\) 3. \(\phi = \cos^{-1} \left(\frac{z}{\rho}\right)\) Based on these formulas, calculate \(\rho\), \(\theta\), and \(\phi\) using \(x = 4\), \(y = 1\), and \(z = -5\). Fill in the boxes with the calculated values for \(\rho\), \(\theta\), and \(\phi\). Feel free to use a calculator for finding square roots and inverse trigonometric functions as needed.
Transcribed Image Text:**Question:** What are the spherical coordinates of the point whose rectangular coordinates are \((4, 1, -5)\)? **Solution:** To find the spherical coordinates \((\rho, \theta, \phi)\), we use the following conversions from rectangular coordinates \((x, y, z)\): 1. \(\rho = \sqrt{x^2 + y^2 + z^2}\) 2. \(\theta = \tan^{-1} \left(\frac{y}{x}\right)\) 3. \(\phi = \cos^{-1} \left(\frac{z}{\rho}\right)\) Based on these formulas, calculate \(\rho\), \(\theta\), and \(\phi\) using \(x = 4\), \(y = 1\), and \(z = -5\). Fill in the boxes with the calculated values for \(\rho\), \(\theta\), and \(\phi\). Feel free to use a calculator for finding square roots and inverse trigonometric functions as needed.
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