Problem 50 - Calculus Early Transcendentals

Question

A Cartesian and a polar graph of \(r=f(\theta)\) are given in the figures. Mark the points on the polar graph that correspond to the points shown on the Cartesian graph. \(r=\sin (1+3 \cos \theta)\)

Step-by-Step Solution

Answer

Question: Given a Cartesian graph and a polar graph for the function \(r=\sin (1+3 \cos \theta)\), convert the points from the Cartesian graph to polar coordinates and mark the corresponding points on the polar graph. Explain the process of performing this conversion and identifying the points on both graphs.

Step 1: Find the points on the Cartesian graph

First, identify the points on the Cartesian graph of the function. You will need their coordinates (x, y) to later convert them into polar coordinates.

Step 2: Convert the Cartesian points to polar coordinates

To convert a point (x, y) from Cartesian to polar coordinates, we use the following formulas: \(r=\sqrt{x^2+y^2}\) \(\theta=\arctan\frac{y}{x}\) (with the quadrant's specific considerations for the value of x and y) Apply these formulas to each point found on the Cartesian graph in Step 1.

Step 3: Match the converted points to the polar graph

After obtaining the polar coordinates of the points, check the polar graph and look for their corresponding locations. Keep in mind that a polar coordinate (r, θ) represents a point at a distance r from the origin (pole) and at an angle θ from the polar axis (positive x-axis).

Step 4: Mark the corresponding points on the polar graph

Once you have found the locations of the points on the polar graph, mark them to show their correspondence with the points on the Cartesian graph. The student should now be able to visualize the connection between these two representations of the same function \(r=\sin (1+3 \cos \theta)\).

Key Concepts

Understanding Cartesian Coordinates

Before diving into polar coordinates and conversion, it's essential to understand Cartesian Coordinates. They form the basis of most graphing concepts. The Cartesian coordinate system is composed of two perpendicular axes, usually labeled as the x-axis and y-axis. These axes divide the plane into four quadrants.

• Each point on the graph is represented by an ordered pair \( (x, y) \), where \( x \) is the horizontal position and \( y \) is the vertical position.
• This system is particularly intuitive for positioning points linearly, as one would on graph paper.

Being familiar with Cartesian coordinates is key to understanding the basic layout of graphs before we venture into their polar counterparts. By identifying certain points on a Cartesian graph, you set a foundation to convert these into polar coordinates.

Coordinate Conversion Insights

Converting coordinates is a crucial skill when transitioning between different graphing systems. For our case, we'll convert Cartesian coordinates to polar coordinates.

• Polar coordinates are represented by \( (r, \theta) \), where \( r \) is the distance from the origin (or pole) and \( \theta \) is the angle from the polar axis.
• The conversion involves determining \( r \) using the formula \( r = \sqrt{x^2 + y^2} \).
• To find \( \theta \), use \( \theta = \arctan\frac{y}{x} \), which considers the current quadrant to correctly locate the angle.

This transformation from Cartesian to polar allows you to understand not just where a point is in relation to the origin, but how it behaves within a circular system. Practicing these conversions is essential for graphing complexities.

Graphing Functions with Polar Coordinates

Polar graphs offer an interesting way to visualize functions differently from Cartesian graphs. Let's break it down:

• The function \( r = \sin(1 + 3 \cos \theta) \) is plotted on a polar graph, where \( \theta \) varies, tracing a path with variable radius \( r \).
• This function creates intricate patterns that are more complex than simple lines or curves in Cartesian systems.
• Polar graphs allow us to explore circular symmetries and periodicities that are not as evident in Cartesian graphs.

By understanding these fundamentals of polar graphing, you are equipped to judge how certain points and motions correspond to the transformations observed in Cartesian systems. Identifying and marking corresponding points helps deepen your comprehension of these two diverse but interconnected coordinate systems.