Problem 50 - Calculus Early Transcendentals: Pearson New International Edition

Question

In many cases, polar graphs are related to each other by rotation. We explore that concept here. (a) How are the graphs of \(r=1+\sin (\theta-\pi / 3)\) and \(r=\) \(1+\sin (\theta+\pi / 3)\) related to the graph of \(r=1+\sin \theta\) ? (b) How is the graph of \(r=1+\sin \theta\) related to the graph of \(r=1-\sin \theta\) ? (c) How is the graph of \(r=1+\sin \theta\) related to the graph of \(r=1+\cos \theta\) ? (d) How is the graph of \(r=f(\theta)\) related to the graph of \(r=f(\theta-\alpha)\) ?

Step-by-Step Solution

Answer

(a) Rotated by \( \frac{\pi}{3} \) radians. (b) Reflection over \( \theta = \frac{\pi}{2} \). (c) Rotation by \( \frac{\pi}{2} \) radians. (d) Rotation by \( \alpha \) radians.

Step 1: Understanding Polar Graph Rotations

To solve this problem, understand that rotating a graph involves changing the angle \( \theta \). Rotations are made by adding or subtracting a constant from \( \theta \). For instance, \( r = 1 + \sin(\theta - \alpha) \) rotates the graph of \( r = 1 + \sin(\theta) \) by \( \alpha \) radians clockwise if \( \alpha > 0 \), and counterclockwise if \( \alpha < 0 \).

Step 2: Comparing Graph in Part (a)

For \( r = 1 + \sin(\theta-\frac{\pi}{3}) \) and \( r = 1 + \sin(\theta+\frac{\pi}{3}) \):- \( r = 1 + \sin(\theta-\frac{\pi}{3}) \) is the graph of \( r = 1 + \sin \theta \) rotated \( \frac{\pi}{3} \) radians clockwise.- \( r = 1 + \sin(\theta+\frac{\pi}{3}) \) is the graph of \( r = 1 + \sin \theta \) rotated \( \frac{\pi}{3} \) radians counterclockwise.

Step 3: Comparing Graphs in Part (b)

For \( r = 1 + \sin \theta \), its graph is horizontally reflected over the line \( \theta = \frac{\pi}{2} \) to become the graph of \( r = 1 - \sin \theta \). This reflection changes the direction of the highest and lowest points on the graph.

Step 4: Comparing Graphs in Part (c)

The graph of \( r = 1 + \sin \theta \) is a limaçon, while the graph of \( r = 1 + \cos \theta \) is the same in shape but rotated by \( \frac{\pi}{2} \) radians counterclockwise from the sine version. This rotation changes orientation from vertical to horizontal.

Step 5: General Rotation in Part (d)

The graph of \( r = f(\theta - \alpha) \) is a graph of \( r = f(\theta) \) rotated \( \alpha \) radians clockwise around the origin if \( \alpha > 0 \) and counterclockwise if \( \alpha < 0 \). This is because subtracting \( \alpha \) from \( \theta \) effectively shifts the starting point of the function.

Key Concepts

Graph Rotation

When working with polar coordinates, graph rotation is a key concept. It involves changing the angle, denoted as \( \theta \), to either rotate the graph clockwise or counterclockwise. This is achieved by adding or subtracting a fixed angle, known as the rotation angle (\( \alpha \)), from \( \theta \).
This operation affects the orientation of graphs in polar coordinates, particularly affecting graphs like \( r = 1 + \sin(\theta) \).
For example, to rotate the graph by \( \frac{\pi}{3} \) radians clockwise, we adjust \( \theta \) to \( \theta - \frac{\pi}{3} \). Conversely, for a counterclockwise rotation, \( \theta \) is modified to \( \theta + \frac{\pi}{3} \).
This concept is crucial in comparing various polar graphs by how their orientations differ due to these rotations.

Trigonometric Functions

Trigonometric functions such as sine and cosine play a significant role in polar graphs. They determine the shape and orientation of these graphs.
When examining equations like \( r = 1 + \sin \theta \) and \( r = 1 + \cos \theta \), we notice that sine creates a vertically oriented graph called a limaçon, whereas cosine produces a horizontally oriented one.
This contrast in orientation is due to the inherent properties of sine and cosine functions: sine peaks at \( \theta = \frac{\pi}{2} \) while cosine peaks at \( \theta = 0 \).
Thus, changing the trigonometric function within a polar equation can drastically impact the graph's appearance and orientation.

Polar Graphs

Polar graphs are a visual representation of equations in polar coordinates. These graphs plot points \((r, \theta)\) on a polar plane, where \( r \) is the radial distance from the origin, and \( \theta \) is the angle from the polar axis.
Equations like \( r = 1 + \sin \theta \) form intricate shapes such as limaçons, which have a specific shape dictated by the trigonometric function involved.
An interesting aspect of polar graphs is that rotating them involves simply changing the angle \( \theta \) by an offset \( \alpha \). This simplicity in rotation makes polar graphs a powerful tool in mathematical modeling and visualization.

Graph Reflection

Graph reflection in polar coordinates often results in a visual flipping of the graph across a specific axis. This happens when a function transforms like \( r=1+\sin \theta \) to \( r=1-\sin \theta \).
The graph reflects over the line \( \theta = \frac{\pi}{2} \), essentially inverting its highest and lowest points.
This form of transformation alters the graph's appearance without changing its shape. These changes in visual representation are crucial for understanding symmetries within polar graphs and the effect of transformations on them.