Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 2: Problem 408
For the following exercises, the spherical coordinates of a point are given. Find its associated cylindrical coordinates. \(\left(4, \frac{\pi}{4}, \frac{\pi}{6}\right)\)
Short Answer
Expert verified
Cylindrical coordinates: \((2, \frac{\pi}{4}, 2\sqrt{3})\).
Step by step solution
01
Understand the Problem
We need to convert the spherical coordinates \((4, \frac{\pi}{4}, \frac{\pi}{6})\) into cylindrical coordinates. In spherical coordinates, \(4\) is the radial distance \(\rho\), \(\frac{\pi}{4}\) is the azimuthal angle \(\theta\), and \(\frac{\pi}{6}\) is the polar angle \(\phi\). For cylindrical coordinates, we need to find \((r, \theta, z)\).
02
Calculate the Radial Distance r
In cylindrical coordinates, the radial distance \(r\) is given by \(r = \rho \sin \phi\). Substitute the values: \(\rho = 4\) and \(\phi = \frac{\pi}{6}\). \[ r = 4 \sin \left(\frac{\pi}{6}\right) = 4 \times \frac{1}{2} = 2 \]
03
Determine the Azimuthal Angle 胃
The azimuthal angle \(\theta\) in cylindrical coordinates is the same as the azimuthal angle in spherical coordinates. Therefore, \(\theta = \frac{\pi}{4}\).
04
Find the z-coordinate
The \(z\)-coordinate in cylindrical coordinates is obtained using \(z = \rho \cos \phi\). Substitute the values: \(\rho = 4\) and \(\phi = \frac{\pi}{6}\). \[ z = 4 \cos \left(\frac{\pi}{6}\right) = 4 \times \frac{\sqrt{3}}{2} = 2\sqrt{3} \]
05
Write the Cylindrical Coordinates
Combine the calculated values to write the cylindrical coordinates: \( (r, \theta, z) = (2, \frac{\pi}{4}, 2\sqrt{3}) \).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cylindrical Coordinates
Cylindrical coordinates are a three-dimensional coordinate system that extends polar coordinates by adding a height element. They're called cylindrical because they naturally describe shapes like cylinders. This system is useful for problems involving symmetry around an axis.
- Components: Cylindrical coordinates consist of three elements \(r, \theta, z\)\;, where \(r\) is the radial distance from the origin in the xy-plane, \(\theta\) is the angular displacement from a reference direction, and \(z\) is the height above the xy-plane.
- Applications: It's particularly useful in engineering and physics for modeling problems where one of the axes is rotationally symmetric, like the flow of fluids in a pipe.
Spherical Coordinates
Spherical coordinates offer a powerful way to describe locations in a three-dimensional space, particularly for points on spheres. It's like a globe's latitude and longitude but includes the radius too.
- Components: These coordinates are determined by \(\rho, \theta, \phi\)\; where \(\rho\) is the distance from the origin to the point, \(\theta\) is the angle in the xy-plane from the positive x-axis, and \(\phi\) is the angle from the positive z-axis to the point.
- Conversions: Given their nature, spherical coordinates convert naturally to both Cartesian and cylindrical systems through trigonometric relationships, helping solve spatial problems.
- Usage: They are immensely helpful in physics, particularly in scenarios involving waves and fields where symmetry around a point makes the math simpler.
Trigonometric Functions
Trigonometric functions, like sine and cosine, play a key role in converting between coordinate systems by defining relationships between angles and distances. In coordinate conversion, they're like bridge-builders.
- Sine Function: Used to determine the radial distance \(r\) in cylindrical coordinates from spherical inputs, via the formula \(r = \rho \sin \phi\).
- Cosine Function: Vital for finding the z-coordinate through \(z = \rho \cos \phi\).
- Angles and Distances: Trigonometric functions relate angles to linear dimensions, which is essential for translating between different coordinate systems.
Mathematical Problem Solving
Mathematical problem-solving prowess is key when transitioning between different coordinate systems. It involves analysis, strategy, and computation.
- Understanding the Problem: The initial step in any conversion is recognizing each element of the coordinate systems involved鈥攌nowing what needs mapping to what.
- Strategic Calculations: Carefully apply the appropriate trigonometric formulas to translate between systems, ensuring each step logically follows the previous.
- Verification: Always check the final calculations to ensure accuracy鈥攕mall errors in trigonometric computation can lead to significant deviations.
