Problem 73 - Advanced Engineering Mathematics

Question

In Problems \(71-74\), convert the given equation to rectangular coordinates. $$ \rho=2 \sec \phi $$

Step-by-Step Solution

Answer

The equation is \( z = 2 \) in rectangular coordinates.

Step 1: Identify the given equation

We have the equation in spherical coordinates: \( \rho = 2 \sec \phi \). Our task is to convert this equation into rectangular coordinates \((x, y, z)\).

Step 2: Understand spherical to rectangular conversion

In spherical coordinates, \( \rho \) is the radial distance, \( \phi \) is the azimuthal angle, and \( \theta \) is the polar angle. The conversion formulas are: \[ x = \rho \sin \phi \cos \theta \] \[ y = \rho \sin \phi \sin \theta \] \[ z = \rho \cos \phi \]

Step 3: Use the definition of secant

Recall that \( \sec \phi = \frac{1}{\cos \phi} \). Substitute the definition into the equation \( \rho = 2 \sec \phi \), which gives us \( \rho = \frac{2}{\cos \phi} \).

Step 4: Solve for \( \rho \cos \phi \)

Multiply both sides of \( \rho = \frac{2}{\cos \phi} \) by \( \cos \phi \) to obtain \( \rho \cos \phi = 2 \).

Step 5: Substitute using conversion formulas

From spherical to rectangular we know \( z = \rho \cos \phi \). Thus, replace \( \rho \cos \phi \) from Step 4 with \( z \), resulting in the equation: \[ z = 2 \].

Step 6: Write the final equation

The equation \( z = 2 \) is the representation of the original spherical equation in rectangular coordinates.

Key Concepts

Spherical Coordinates

Spherical coordinates are a system of three-dimensional coordinate geometry. In it, a point in space is expressed with three values: the radial distance \( \rho \), the polar angle \( \phi \), and the azimuthal angle \( \theta \). This system is especially useful in situations where the origin is the natural center of rotation, like celestial systems or physics problems concerning spherical bodies.

Here's a quick overview:

• \( \rho \) is the distance from the origin to the point.
• \( \phi \) is the angle from the positive z-axis down to the point.
• \( \theta \) is the angle from the positive x-axis to the point's projection in the xy-plane.

The spherical coordinate system can help simplify many problems but converting them to another system can be tricky without knowledge of conversion techniques.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are one of the most commonly used coordinate systems. In this system, a position in space is described using the \( x \), \( y \), and \( z \) axes. It's straightforward and often applied in everyday geometry, physics, and engineering.

Some key aspects include:

• Each axis is perpendicular to the others, dividing space into quadrants or octants.
• It's used extensively for graph plotting and in applied sciences.
• Conversion from other coordinate systems often ends in simpler equations results in this form.

Understanding the rectangular system's simplicity and versatility is vital for translating problems across various fields of study.

Coordinate Conversion

Coordinate conversion is the process of translating between different coordinate systems to ease problem-solving based on the scenario. When converting from spherical to rectangular coordinates, we use specific formulas:

• \( x = \rho \sin \phi \cos \theta \)
• \( y = \rho \sin \phi \sin \theta \)
• \( z = \rho \cos \phi \)

In our exercise, we used this understanding to translate \( \rho = 2 \sec \phi \) into a rectangular equation. It involved understanding the definition of secant, and realizing that the radial measure in spherical coordinates, when translated correctly, dictated an equation in the familiar rectangular terms.

Coordinate conversion is crucial for solving complex geometry problems and ensuring measurement precision in diverse scientific applications.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. They play a pivotal role when converting between different coordinate systems like spherical and rectangular.

Consider the secant function: \( \sec \phi = \frac{1}{\cos \phi} \). This identity helped us transform \( \rho = 2 \sec \phi \) into \( \rho = \frac{2}{\cos \phi} \). Knowing this allowed multiplication with \( \cos \phi \) to simplify to \( \rho \cos \phi = 2 \), making it easy to apply the conversion formula directly as \( z = 2 \).

Trigonometric identities streamline the conversion and also enhance understanding of relationships within trigonometric functions. They function as tools simplifying computations in mathematics, physics, and engineering.