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) Rotations of \( \pm \pi/3 \). (b) Reflection across x-axis. (c) \( \pi/2 \) phase shift. (d) Rotation by \( +\alpha \).

Step 1: Interpret Rotation in Polar Graphs

Polar graphs of the form \( r = 1 + \sin(\theta - \alpha) \) are rotations of \( r = 1 + \sin\theta \). Moving \( \theta \) by \( +\alpha \) in the equation rotates the graph by \( -\alpha \) on the polar plane, while moving \( \theta \) by \( -\alpha \) rotates it by \( +\alpha \).

Step 2: Relation in Part (a): Rotation by \( \pm \pi/3 \)

The graph of \( r = 1 + \sin(\theta - \pi/3) \) is a rotation of \( r = 1 + \sin\theta \) by \( +\pi/3 \) (counter-clockwise), and \( r = 1 + \sin(\theta + \pi/3) \) is a rotation of \( r = 1 + \sin\theta \) by \( -\pi/3 \) (clockwise).

Step 3: Relation in Part (b): Reflection Across x-Axis

The graph of \( r = 1 + \sin\theta \) is reflected across the x-axis to obtain \( r = 1 - \sin\theta \). Reflection in polar coordinates can be achieved by switching the sign of the sine term.

Step 4: Relation in Part (c): Phase Shift by \( \pi/2 \)

The graph of \( r = 1 + \sin\theta \) is equivalent to the graph of \( r = 1 + \cos(\theta) \) shifted \( \pi/2 \) counter-clockwise. This arises because \( \cos\theta = \sin(\theta + \pi/2) \).

Step 5: General Relationship in Part (d)

The graph of \( r = f(\theta - \alpha) \) is the graph of \( r = f(\theta) \) rotated by \( +\alpha \) counter-clockwise. This is a standard property of rotational transformations in polar coordinates.

Key Concepts

Rotation in Polar Coordinates

Polar graphs often involve rotations, allowing us to transform a graph's orientation without changing its shape. When you have a polar equation like \( r = 1 + \sin(\theta - \alpha) \), the graph rotates by an angle \( \alpha \) counter-clockwise from the graph of \( r = 1 + \sin\theta \). Conversely, if the equation is \( r = 1 + \sin(\theta + \alpha) \), the graph rotates clockwise by \( \alpha \). This means:

• Subtracting \( \pi/3 \) from \( \theta \) rotates the graph by \( +\pi/3 \) (counter-clockwise).
• Adding \( \pi/3 \) to \( \theta \) results in a \( -\pi/3 \) rotation (clockwise).

Understanding these concepts helps in predicting how a graph will look and behave after a rotation, making problems like those in the exercise easier to tackle.

Reflection in Polar Graphs

Reflection in polar graphs involves flipping a graph over a line, usually the x-axis or y-axis. For example, the equation \( r = 1 + \sin\theta \) becomes \( r = 1 - \sin\theta \) when reflected across the x-axis. This transformation results in:

• The peak points of the graph getting mirrored across the x-axis.
• The general symmetry of the graph being maintained but inverted.

To reflect a graph in polar coordinates, you usually modify the trigonometric term within the equation. Understanding reflections allows you to visualize and generate symmetric shapes effortlessly.

Phase Shift in Trigonometric Functions

The concept of phase shift helps us understand the horizontal shifting of graphs formed by trigonometric functions. In polar coordinates, a phase shift transforms, for example, \( r = 1 + \sin\theta \) to \( r = 1 + \cos\theta \). Since \( \cos\theta = \sin(\theta + \pi/2) \), this transformation indicates a \( \pi/2 \) shift counter-clockwise. Highlights of phase shifts include:

• Enabling the alignment of trigonometric functions in different orientations.
• Maintaining the overall shape but altering the angle start position.

By comprehending phase shifts, you can manipulate the starting position of cycles, which is particularly helpful in signal processing and wave analysis.

Polar Coordinates Transformation

Polar coordinates transformation involves converting between different forms of polar equations or changing the perspective of the graph. This concept is evident in the transformation \( r = f(\theta - \alpha) \), where the graph shifts counter-clockwise by \( \alpha \) degrees. Key points about polar transformations include:

• Rotational transformations are pivotal for aligning graphs in desired orientations.
• They provide a flexible approach to analyze rotational symmetries within graphs.

Transformations in polar coordinates are a powerful tool for modifying the presentation of equations, allowing for a cohesive analysis of the orientation and symmetry in polar graphs.