Problem 31 - Thomas Calculus

Question

Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian equations. Then describe or identify the graph. $$r=4 \csc \theta$$

Step-by-Step Solution

Answer

The graph is a horizontal line at \(y = 4\).

Step 1: Understand the Polar Equation

The given polar equation is \( r = 4 \csc \theta \). The cosecant function is the reciprocal of sine, so we can rewrite \( \csc \theta \) as \( \frac{1}{\sin \theta} \). Therefore, the equation becomes \( r = \frac{4}{\sin \theta} \).

Step 2: Rewrite in Terms of Cartesian Coordinates

Recall the polar to Cartesian conversion formulas: \( x = r \cos \theta \), \( y = r \sin \theta \), and \( r^2 = x^2 + y^2 \). From \( \sin \theta = \frac{y}{r} \), substitute \( r \) in the equation \( r = \frac{4}{\sin \theta} \): \( r = \frac{4r}{y} \).

Step 3: Simplify the Equation

Multiply both sides by \( y \) to eliminate the fraction: \( y = 4 \).

Step 4: Identify the Graph

The Cartesian equation \( y = 4 \) is a horizontal line that crosses the y-axis at \( (0, 4) \) and extends indefinitely in both the positive and negative x-directions.

Key Concepts

Polar Coordinates

Polar coordinates are a unique way to locate points in a plane. Unlike Cartesian coordinates that use an X-axis and a Y-axis to define points, polar coordinates use a radius and an angle. The radius, often denoted by the letter \( r \), represents how far away the point is from the origin, which is the center of the plane. The angle, \( \theta \), shows how much to rotate from the positive X-axis. Consider the pole as the origin of the plane and the polar axis as the starting line for the angle.
Polar coordinates are very useful in scenarios where circular or rotational motion is involved. For example, polar coordinates are often employed in engineering, physics, and navigation to describe the positions of objects around a central point.

• The polar coordinate system is defined as \( (r, \theta) \).
• Converting between polar and Cartesian coordinates allows for different perspectives and solutions to problems.

Cartesian Coordinates

In the Cartesian coordinate system, each point is determined by an ordered pair of numbers \((x, y)\). These numbers represent distances along two perpendicular axes: the X-axis (horizontal) and the Y-axis (vertical). This system is the most commonly used, especially in basic math and geometry, due to its straightforward nature.
The transition from polar to Cartesian coordinates involves a few standard relationships:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)
• \( r^2 = x^2 + y^2 \)

These equations help to convert the circular properties of polar coordinates to the linear layout of Cartesian coordinates. For example, in the given problem, we transform the polar equation \( r = 4 \csc \theta \) into the Cartesian equation \( y = 4 \), describing a simple horizontal line in the Cartesian plane. The conversion is essential in mathematics as it allows tackling problems from a different angle, providing clarity and simplicity.

Graphing Equations

Graphing equations involves plotting the result of equations on a coordinate system. With Cartesian coordinates, lines are usually easy to plot, as they often follow simple linear rules, like \( y = mx + b \). In the exercise example, the equation \( y = 4 \) describes a horizontal line crossing the Y-axis at 4. This horizontal line covers all points where Y equals 4, irrespective of the X-value.
However, polar graphs feature curves and circles more frequently than straight lines. For instance, if our initial equation were plotted in polar coordinates, it might form a different shape. By converting polar equations to Cartesian form, students can identify known shapes like lines, circles, or parabolas.
When graphing:

• Ensure the equation is in a recognizable form. For Cartesian equations, this typically involves simplifying to the \( y = mx + b \) form for linear equations.
• Identify key features like intercepts, slopes, and symmetries.
• Use plotting points to ensure accuracy if the equation doesn't fit a common template like linear or quadratic forms.

By converting and graphing equations, students gain insight into the behavior and form of mathematical functions, enhancing their analytical skills.