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Chapter 2: Problem 405

For the following exercises, the cylindrical coordinates of a point are given. Find its associated spherical coordinates, with the measure of the angle \(\varphi\) in radians rounded to four decimal places. \(\left(3, \frac{\pi}{2}, 3\right)\)

Short Answer

Expert verified
The spherical coordinates are \(\left(3\sqrt{2}, \frac{\pi}{2}, \frac{\pi}{4}\right)\).

Step by step solution

01

Understand Cylindrical Coordinates

In cylindrical coordinates, the point is given as \((r, \theta, z) = \left(3, \frac{\pi}{2}, 3\right)\). Here, \(r\) is the radial distance from the \(z\)-axis, \(\theta\) is the angle in the \(xy\)-plane from the positive \(x\)-axis, and \(z\) is the height above the \(xy\)-plane.
02

Convert to Spherical Radius \(\rho\)

The spherical radial coordinate \(\rho\) is calculated using the formula \(\rho = \sqrt{r^2 + z^2}\). Substituting the given values, we get:\[ \rho = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \].
03

Determine Spherical Angle \(\phi\)

The spherical angle \(\phi\) is the angle from the positive \(z\)-axis, calculated using the formula \(\tan \phi = \frac{r}{z}\). Substituting the given values, we find:\[ \tan \phi = \frac{3}{3} = 1 \].Thus, \(\phi = \frac{\pi}{4}\).
04

Spherical Angle \(\theta\)

In spherical coordinates, the angle \(\theta\) remains the same as in cylindrical coordinates. Therefore, \(\theta = \frac{\pi}{2}\).
05

Write Spherical Coordinates

The spherical coordinates \((\rho, \theta, \phi)\) are given by \(\left(3\sqrt{2}, \frac{\pi}{2}, \frac{\pi}{4}\right)\), where \(\rho\) is the radial distance, \(\theta\) is the azimuthal angle, and \(\phi\) is the polar angle.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cylindrical Coordinates
Cylindrical coordinates are an extension of two-dimensional polar coordinates. They add a third dimension, making them suitable for representing points in three-dimensional space. In this coordinate system, each point is described by three values:
  • **r (radial distance):** This is the distance from the point to the z-axis. In the given example, it's 3.
  • **胃 (azimuthal angle):** This represents the angle in the xy-plane, measured from the positive x-axis. In our case, this is \(\frac{\pi}{2}\).
  • **z (height):** This is the height of the point above the xy-plane. Here, it's 3.
Cylindrical coordinates are especially useful in scenarios involving circular or rotational symmetry, often simplifying the description of such objects.
Coordinate Conversion
When converting from cylindrical to spherical coordinates, we change from \((r, \theta, z)\) to \((\rho, \theta, \phi)\). Here鈥檚 how each component transforms:
  • **From r to \(\rho\):** The spherical radial distance \(\rho\) is found using \( \rho = \sqrt{r^2 + z^2} \). In the exercise, this calculation gives us \( 3 \sqrt{2} \).
  • **The angle \(\theta\):** It remains unchanged in conversion from cylindrical to spherical. Thus, \(\theta\) stays as \(\frac{\pi}{2}\).
  • **From z to \(\phi\):** The angle from the positive z-axis \(\phi\) is determined by \( \tan(\phi) = \frac{r}{z} \). This calculation results in \( \phi = \frac{\pi}{4} \).
Using this conversion process ensures that each point is accurately represented in spherical coordinates.
Trigonometry
Trigonometry plays a crucial role in coordinate conversions, especially when transforming angles and distances. Understanding the relationships between angles and sides in triangles is key. Here are some important aspects:
  • **Pythagorean Theorem:** Essential for calculating the spherical radius \(\rho\), given by \( \rho = \sqrt{r^2 + z^2} \).
  • **Tangent Function:** To find the angle \(\phi\), the tangent function is employed, \( \tan(\phi) = \frac{r}{z} \). This helps translate vertical height and radial distance into a findable angle from the positive z-axis.
Trigonometry allows us to bridge the gap between different coordinate systems effectively and precisely.
Radians Calculation
Radians are a measure of angle used widely in mathematics and physics, especially in trigonometry and coordinate systems. They provide a direct relationship between the angle and the arc length. When working with these problems:
  • **Defining Radians:** One radian corresponds to an angle whose arc length is equal to the radius of the circle. The complete circle measures \(2\pi\) radians.
  • **Conversions:** Commonly needed when converting degrees to radians (e.g., 180掳 equals \(\pi\) radians).
  • **In Practice:** In this exercise, recognizing \(\frac{\pi}{2}\) signifies a 90掳 turn, and \(\frac{\pi}{4}\) corresponds to a 45掳 angle can illuminate understanding.
Mastering radians helps simplify many mathematical operations involving angles, enabling more efficient calculations and deeper comprehension of the geometry involved.

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Most popular questions from this chapter

For the following exercises, the rectangular coordinates \((x, y, z)\) of a point are given. Find the spherical coordinates \((\rho, \theta, \varphi)\) of the point. Express the measure of the angles in degrees rounded to the nearest integer. \((-2,2 \sqrt{3}, 4)\)

For the following exercises, the equation of a quadric surface is given. a. Use the method of completing the square to write the equation in standard form. b. Identify the surface. \(x^{2}+z^{2}-4 y+4=0\)

For the following exercises, the equation of a surface in spherical coordinates is given. Find the equation of the surface in rectangular coordinates. Identify and graph the surface. \([\mathrm{T}] \rho=4 \csc \varphi\)

For the following exercises, the equation of a quadric surface is given. a. Use the method of completing the square to write the equation in standard form. b. Identify the surface. \(x^{2}+\frac{y^{2}}{4}-\frac{z^{2}}{3}+6 x+9=0\)

For the following exercises, the spherical coordinates of a point are given. Find its associated cylindrical coordinates. \(\left(9,-\frac{\pi}{6}, \frac{\pi}{3}\right)\)
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