Problem 20 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the Cartesian equations of the graphs of the given polar equations. $$ r-5 \cos \theta=0 $$

Step-by-Step Solution

Answer

The Cartesian equation is \( x^3 + xy^2 = 5x^2 \).

Step 1: Understand the Problem

We are given the polar equation \( r - 5 \cos \theta = 0 \) and need to convert it into a Cartesian equation. In polar coordinates, \( r \) is the radius (distance from the origin) and \( \theta \) is the angle from the positive x-axis.

Step 2: Express \( r \) in Terms of \( \cos \theta \)

Solve the given polar equation for \( r \):\[ r - 5 \cos \theta = 0 \ \Rightarrow r = 5 \cos \theta\]

Step 3: Convert Polar Coordinates to Cartesian Coordinates

Use the polar to Cartesian conversion formulas: \( x = r \cos \theta \) and \( y = r \sin \theta \). Also, \( r^2 = x^2 + y^2 \).Since \( r = 5 \cos \theta \), we can substitute \( r \) into the first formula: \[ x = r \cos \theta = (5 \cos \theta) \cos \theta = 5 \cos^2 \theta\]

Step 4: Relate \( \cos \theta \) to Cartesian Coordinates

Using the relation \( \cos \theta = \frac{x}{r} \):\[ \cos^2 \theta = \left(\frac{x}{r}\right)^2 = \frac{x^2}{r^2}\]

Step 5: Substitute Back and Simplify

Replace \( \cos^2 \theta \) in the expression for \( x \) found earlier:\[ x = 5 \cdot \frac{x^2}{r^2}\]Since \( r^2 = x^2 + y^2 \), substitute to obtain:\[ x = 5 \cdot \frac{x^2}{x^2 + y^2}\]

Step 6: Derive the Cartesian Equation

Clear the fraction by multiplying both sides by \( x^2 + y^2 \):\[ x(x^2 + y^2) = 5x^2\]Rearrange to get:\[ x^3 + xy^2 = 5x^2\]This is the Cartesian equation of the given polar equation.

Key Concepts

Cartesian coordinates

Cartesian coordinates are used to specify the position of points in a plane using a pair of numerical values. These values, \(x\) and \(y\), represent positions on the horizontal and vertical axes, respectively. The Cartesian coordinate system is fundamentally a grid system that allows you to pinpoint any location in a 2D space by simply giving two numbers.

• The first number in a Cartesian coordinate is called the x-coordinate. It indicates how far left or right a point is from the origin (0,0).
• The second number is the y-coordinate, which shows how far up or down the point is from the origin.

The system is very intuitive and extremely useful in mathematics and engineering for mapping spaces and functions easily. This method is especially handy when dealing with functions and graphs, as it allows one to visualize algebraic equations.

polar coordinates

Polar coordinates offer an alternative method for describing the position of points using a radius and an angle. Instead of x and y, these coordinates use \(r\) and \(\theta\).

• \(r\) is the radial distance from the origin to the point.
• \(\theta\) is the angle between the positive x-axis and the line segment connecting the origin to the point, measured counterclockwise.

Polar coordinates are particularly useful in scenarios involving circular or rotational symmetry, as they simplify the equations and make angular movements easier to understand.
For instance, when graphing certain types of curves, like spirals or circles, using polar coordinates can be more straightforward than using Cartesian coordinates.

conversion formulas

Converting between polar and Cartesian coordinates involves a set of important formulas that bridge these two systems. For anyone frequently switching between these coordinate systems, it's essential to memorize these conversions.
Here are the main conversion formulas:

• To convert a point from polar to Cartesian:
  • \(x = r \cos \theta\)
  • \(y = r \sin \theta\)
• To convert a point from Cartesian to polar:
  • \(r = \sqrt{x^2 + y^2}\)
  • \(\theta = \tan^{-1}(\frac{y}{x})\)

To solve problems that require conversions, such as the one provided in the exercise, knowing these formulas helps switch the format of the equations, revealing insights into the relationship between the coordinates.

trigonometric functions

Trigonometric functions play a crucial role in converting polar coordinates to Cartesian coordinates. They allow the conversion formulas to express relationships in either rectangular or polar form by leaning on the circle's properties.
Key trigonometric functions include:

• Cosine (\(\cos\)) - Describes the horizontal component relative to an angle in the circle.
• Sine (\(\sin\)) - Represents the vertical component in the context of an angle.
• Tangent (\(\tan\)) - A ratio of the sine and cosine, often used for finding angles in coordinate transformations.

In mathematics, these functions are fundamental for solving problems involving angles and lengths. When addressing exercises like converting from polar to Cartesian, \(\cos \, \text{and}\, \sin\) are directly employed to map the radial and angular descriptions into x and y coordinates.