Problem 65 - Advanced Engineering Mathematics

Question

In Problems \(63-66\), convert the points given in rectangular coordinates to spherical coordinates. $$ \left(\frac{\sqrt{3}}{2}, \frac{1}{2}, 1\right) $$

Step-by-Step Solution

Answer

The spherical coordinates are \(\left(\sqrt{2}, \frac{\pi}{6}, \frac{\pi}{4}\right)\).

Step 1: Understand Rectangular to Spherical Conversion

To convert from rectangular coordinates \(x, y, z\) to spherical coordinates \(\rho, \theta, \phi\), use the following formulas: \(\rho = \sqrt{x^2 + y^2 + z^2}\), \(\theta = \tan^{-1}(\frac{y}{x})\), and \(\phi = \cos^{-1}(\frac{z}{\rho})\).

Step 2: Calculate \(\rho\) (The Radius)

First, calculate the radius \(\rho\) using the formula \(\rho = \sqrt{x^2 + y^2 + z^2}\). Substitute the given values for x, y, and z:\(\rho = \sqrt{\left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{1}{2}\right)^2 + (1)^2} = \sqrt{\frac{3}{4} + \frac{1}{4} + 1} = \sqrt{2}\)

Step 3: Calculate \(\theta\) (The Azimuthal Angle)

Next, calculate the angle \(\theta\) using the formula \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\). Substitute the values of x and y:\(\theta = \tan^{-1}\left(\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}\right) = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6}\)

Step 4: Calculate \(\phi\) (The Polar Angle)

Finally, compute the angle \(\phi\) using the formula \(\phi = \cos^{-1}(\frac{z}{\rho})\). Using the calculated radius and given value for z:\(\phi = \cos^{-1}\left(\frac{1}{\sqrt{2}}\right) = \frac{\pi}{4}\)

Step 5: Final Result: Combine Spherical Coordinates

Combine the calculated components to find the spherical coordinates: \(\left(\rho, \theta, \phi\right) = \left(\sqrt{2}, \frac{\pi}{6}, \frac{\pi}{4}\right)\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates are one of the most common coordinate systems used in mathematics and physics. In this system, any point in space is described by three values:

• **X-coordinate**: Measures the point's horizontal distance in the x-direction.
• **Y-coordinate**: Represents the vertical position along the y-direction.
• **Z-coordinate**: Indicates how high or low the point is along the z-axis.

Rectangular coordinates are also known as Cartesian coordinates, named after the mathematician René Descartes. They are very useful when dealing with problems related to dimensions, shapes, and positions in a straight-line relationship. For example, the point \(\left(\frac{\sqrt{3}}{2}, \frac{1}{2}, 1\right)\), tells us exactly how far along each axis the point is located. Each coordinate provides unique data about the point's position, making it easier to visualize and calculate distances.

Coordinate Transformation

Coordinate transformation is the process of converting points from one coordinate system to another, such as from rectangular to spherical coordinates. Such transformations are vital for solving problems where one system of coordinates is more convenient than another. For conversions like arithmetic and trigonometric calculations, spherical coordinates often make 3D problems simpler. The formulas used in transforming from rectangular \((x,y,z)\) to spherical coordinates \((\rho, \theta, \phi)\) are:

• **\(\rho\)**: The radial distance. Calculate by \(\rho = \sqrt{x^2 + y^2 + z^2}\), indicating the point's distance from the origin.
• **\(\theta\)**: The azimuthal angle (in the x-y plane), found using \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\).
• **\(\phi\)**: The polar angle (angle from the z-axis), calculated through \(\phi = \cos^{-1}\left(\frac{z}{\rho}\right)\).

These transformations can shed light on the physical interpretation of a point's position more clearly when considering spheres, circles, or rotational motion.

Trigonometric Functions

Trigonometric functions play a crucial role in converting between various coordinate systems, notably between rectangular and spherical coordinates. They help us understand angles and distances between points relative to specific axes. In the process of coordinate transformation, two trigonometric functions are essential:

• **Tangent Function**: Used to find the angle \(\theta\) in the x-y plane. This helps determine the direction of the point from the origin.
• **Inverse Tangent**: Calculated by its formula \(\tan^{-1}\left(\frac{y}{x}\right)\), gives the azimuthal angle \(\theta\), providing insight into the radial alignment in horizontal planes.
• **Cosine Function**: Applied to find the angle \(\phi\). It measures the deviation from the z-axis and is calculated using \(\cos^{-1}\left(\frac{z}{\rho}\right)\).

Understanding these functions aids in translating between coordinate systems while preserving geometric and spatial relationships, making spatial problems more tractable.