Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

Make the required change in the given equation. \(x^{2}+y^{2}=9\) to cylindrical coordinates

Step-by-Step Solution

Answer

The equation in cylindrical coordinates is \\(r^2 = 9\\).

Step 1: Understand the Equation

The given equation is a circle with a radius of 3 in the Cartesian coordinate system. In cylindrical coordinates, which consist of \(r, \, \,\theta\), we need to convert each variable appropriately.

Step 2: Recall Conversion Formulas

The relationship between Cartesian coordinates \(x, y\) and cylindrical coordinates \(r, \, \,\theta\) is defined by the following formula: \(x = r \cos\theta\) and \(y = r \sin\theta\).

Step 3: Substitute Cartesian Variables

Substitute \(x = r \cos\theta\) and \(y = r \sin\theta\) into the equation \(x^{2} + y^{2} = 9\), giving us \( (r \cos\theta)^{2} + (r \sin\theta)^{2} = 9 \).

Step 4: Simplify the Equation

Simplify the equation: \((r \cos\theta)^{2} + (r \sin\theta)^{2} = r^{2} \cos^{2}\theta + r^{2} \sin^{2}\theta = r^{2}(\cos^{2}\theta + \sin^{2}\theta) \).

Step 5: Apply the Pythagorean Identity

Recall the identity \(\cos^{2}\theta + \sin^{2}\theta = 1\). Use this identity to simplify the expression to \(r^{2} \times 1 = r^{2}\).

Step 6: Final Cylindrical Equation

The simplified equation \(r^{2} = 9\) represents the original circle equation in cylindrical coordinates.

Key Concepts

Coordinate Systems

In mathematics, a coordinate system is a framework used to uniquely determine the position of points in a space. The Cartesian coordinate system is one of the most widely known systems, defining positions in 2D using the variables \(x\) and \(y\). This is ideal for grid-like spacings and typical everyday scenarios. However, when dealing with circular shapes or rotational symmetries, cylindrical coordinates often simplify problems. This system uses three variables: \(r\) (the radial distance from the origin), \(\theta\) (the angular position), and \(z\) (vertical position, though it wasn't needed in our current example). In our situation, transforming the circle equation into cylindrical coordinates helps in understanding circular features better.

Conversion Formulas

To move between coordinate systems seamlessly, conversion formulas are essential. Think of them as the bridges between two languages, allowing us to translate descriptions of positions accurately. For Cartesian to cylindrical conversion, we need two main formulas:

• \(x = r \cos\theta\)
• \(y = r \sin\theta\)

Using these, any point found in a Cartesian framework can be translated into cylindrical coordinates. In the exercise, substituting these formulations for \(x\) and \(y\) in the equation allows us to reframe the entire problem into cylindrical terms, making the equation align better with our rotational geometry.

Pythagorean Identity

The Pythagorean identity is a cornerstone of trigonometry. It states that for any angle \(\theta\), \(\cos^{2}\theta + \sin^{2}\theta = 1\). This relationship is particularly helpful in simplifying expressions involving trigonometric functions, especially when those expressions form parts of broader equations. In our problem, this identity was a key step to reduce the complexity of the equation once we substituted the conversion formulas. By acknowledging that the sum of these squared trigonometric functions is always one, we drastically simplified the equation, focusing on the radial component \(r\).

Simplifying Equations

Simplifying equations is about breaking them down to their cleanest, most understandable form. It often involves identifying identities or common factors that make the equation easier to understand or solve. In our problem, after substituting the Cartesian variables with cylindrical ones, we arrived at a rather complicated expression. However, recognizing elements like the Pythagorean identity allowed us to vastly simplify it to \(r^2 = 9\). Such reductions not only make the mathematics easier to handle but also make the geometric interpretation – in this case, a circle of radius 3 – more straightforward.