Problem 21 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 17-22, find the Cartesian equations of the graphs of the given polar equations. \(r \sin \theta-1=0\)

Step-by-Step Solution

Answer

The Cartesian equation is \(y = 1\).

Step 1: Convert Polar to Cartesian Equations

The given polar equation is \(r \sin \theta - 1 = 0\). In polar coordinates, \(x = r \cos \theta\) and \(y = r \sin \theta\). We know that \(r \sin \theta = y\). Substitute this into the equation to get \(y - 1 = 0\).

Step 2: Solve for Cartesian Equation

The equation \(y - 1 = 0\) simplifies to \(y = 1\). This is the Cartesian equation of the given polar equation.

Key Concepts

Understanding Polar Coordinates

Polar coordinates are a way of describing the position of a point in a plane using a distance and an angle. It is different from the typical way we've learned, which uses x and y coordinates. Instead of saying "how far to the left or right" (x-axis) and "how far up or down" (y-axis), polar coordinates use:

• The radius \( r \), which tells us how far the point is from the origin (like the center of a circle).
• The angle \( \theta \), which tells us the direction from the origin to the point. It's usually measured in radians or degrees from the positive x-axis.

For example, a point with coordinates \((r, \theta)\) describes a unique position in a plane. Polar coordinates are especially useful in scenarios involving circular or rotational motion as they simplify the mathematics of rotational problems.
When working with polar equations, you'll often come across forms like \( r = f(\theta) \), which provides a relationship between the radius and the angle. Moving between cartesian and polar coordinates sometimes makes solving particular math problems more convenient.

Converting Equations between Polar and Cartesian Systems

The conversion between polar and cartesian coordinates is essential when different forms of equations are given. Each system has its own advantages, depending on the problem type and context. The relationship between polar and cartesian coordinates can be summarized using the following:

• The x-coordinate in the cartesian plane can be expressed as \( x = r \cos \theta \).
  
• The y-coordinate is given by \( y = r \sin \theta \).
• The radius \( r \) can be found using \( r = \sqrt{x^2 + y^2} \).
• The angle can be calculated using \( \theta = \tan^{-1}(\frac{y}{x}) \).

Taking the formula \( r \sin \theta - 1 = 0 \), we can convert it by substituting \( r \sin \theta \) with \( y \) to get \( y - 1 = 0 \). Thus, its cartesian equivalent is \( y = 1 \). The process often simplifies expressions by allowing us to switch contexts using these relationships.

Graphing Equations using Different Coordinate Systems

Once equations are converted to a usable form, graphing becomes straightforward. In our example, after conversion, we find the equation \( y = 1 \).
This is remarkably simple to visualize in the cartesian plane, as it is a horizontal line at \( y = 1 \) which stretches infinitely to the left and right.

• In a polar context, equations may describe circles, or portions of curves, making them complex to draw by hand.
• Using a cartesian equation provides clarity and precision, especially for linear graphs.

Understanding how to graph requires recognizing the type of curve or line the equation describes, whether it's a line, circle, or another shape based on its form. The cartesian representation made it easier to see this, as it pointed out precisely where our line was located on the plane. When graphed, these converted equations help build intuitive understandings of how data behaves or how shapes relate to one another. It simplifies the intricate details that might be cumbersome if only polar data was used.