Problem 18 - Calculus Early Transcendentals: Pearson New International Edition

Question

Make the required change in the given equation. \(x^{2}-y^{2}=25\) to cylindrical coordinates

Step-by-Step Solution

Answer

In cylindrical coordinates: \\(r^2\cos(2\theta) = 25\\).

Step 1: Understand Cylindrical Coordinates

Cylindrical coordinates \(r, \, \theta, \, z\) are related to Cartesian coordinates \(x, \, y, \, z\) as follows: \(x = r\cos\theta\), \(y = r\sin\theta\), and \(z = z\). We express the given equation in terms of \(r, \, \theta, \, z\).

Step 2: Rewrite the Equation

Given equation is \(x^2 - y^2 = 25\). Substitute \(x = r\cos\theta\) and \(y = r\sin\theta\) to obtain: \((r\cos\theta)^2 - (r\sin\theta)^2 = 25\).

Step 3: Simplify the Equation

Simplify the equation: \((r^2\cos^2\theta) - (r^2\sin^2\theta) = 25\). Factor out \(r^2\) to get \(r^2(\cos^2\theta - \sin^2\theta) = 25\).

Step 4: Use Trigonometric Identity

Utilize the trigonometric identity \((\cos^2\theta - \sin^2\theta) = \cos(2\theta)\). The equation updates to \(r^2\cos(2\theta) = 25\).

Step 5: Final Cylindrical Form

The equation \(r^2\cos(2\theta) = 25\) is the expression in cylindrical coordinates, showing a relationship involving \(r\) and \(\theta\).

Key Concepts

Trigonometric Identities

Trigonometric identities are crucial tools in mathematics that allow us to express relationships between different trigonometric functions. These identities can simplify complex expressions or transform equations into more meaningful forms. In this exercise, we utilized the identity involving squares of cosine and sine:
\[ \cos^2\theta - \sin^2\theta = \cos 2\theta \]
This identity stems from the difference of squares, and it's very useful for combining terms, especially when working with equations in different coordinate systems such as converting Cartesian equations into cylindrical ones. Recognizing and applying these identities can drastically reduce the complexity of a problem and reveal underlying forms that are easier to interpret and solve.

Coordinate Transformation

Coordinate transformation is a mathematical method that lets us convert equations from one coordinate system to another. This is particularly important in physics and engineering, where it can be more convenient to use a coordinate system that simplifies problem geometry.

• Cartesian Coordinates: Uses \((x, y, z)\) and is best for rectangular geometries.
• Cylindrical Coordinates: Uses \((r, \theta, z)\) and is excellent for objects with cylindrical symmetry.

In the given exercise, we transformed the Cartesian equation \(x^{2} - y^{2} = 25\) into cylindrical coordinates. This involves substituting \(x = r \cos \theta\) and \(y = r \sin \theta\). Such transformations leverage the symmetry of the problem, often simplifying the mathematics involved.

Cartesian Coordinates

Cartesian coordinates are the most familiar coordinate system for most students, as they describe a point in space using its horizontal and vertical distances from a set origin. The Cartesian system uses three mutually perpendicular axes, labeled as \(x\), \(y\), and \(z\) in three-dimensional space.

• The \(x\)-coordinate measures horizontal position.
• The \(y\)-coordinate measures vertical position.
• The \(z\)-coordinate measures depth when extended into 3D.

In our exercise, the given equation \(x^{2} - y^{2} = 25\) was initially in Cartesian coordinates. While each point on such an equation in 2D can be easily plotted using these variable values, converting into cylindrical coordinates offers alternative insights, especially for problems where radial symmetry around an axis is involved.