Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

Plot the points whose polar coordinates are \(\left(3, \frac{9}{4} \pi\right)\), \(\left(-2, \frac{1}{2} \pi\right), \quad\left(-2,-\frac{1}{3} \pi\right), \quad(-1,-1), \quad(1,-7 \pi), \quad\left(-3,-\frac{1}{6} \pi\right)\), \(\left(-2,-\frac{1}{2} \pi\right)\), and \(\left(3,-\frac{33}{2} \pi\right)\).

Step-by-Step Solution

Answer

Plot the converted points in Cartesian coordinates after normalizing angles and adjusting for negative radii.

Step 1: Understanding Polar Coordinates

Polar coordinates consist of a radius and an angle, denoted as \((r, \theta)\). The radius \(r\) measures the distance from the origin, while the angle \(\theta\) indicates the direction from the positive x-axis.

Step 2: Converting Polar to Cartesian Coordinates

To plot in a Cartesian plane, convert each polar coordinate \((r, \theta)\) to Cartesian coordinates \((x, y)\) using the formulas: \[x = r \cos(\theta)\] \[y = r \sin(\theta)\]

Step 3: Plotting Points One by One

- For \((3, \frac{9}{4}\pi)\): The angle \(\frac{9}{4}\pi\) is equivalent to \(\frac{3}{4}\pi\). Convert to Cartesian as: \(x = 3 \cos\left(\frac{3}{4}\pi\right)\), \(y = 3 \sin\left(\frac{3}{4}\pi\right)\).- Continuing for each point similarly, remember to adjust angles greater than \(2\pi\) or less than \(-2\pi\) within the standard range \([0, 2\pi]\).- Each coordinate converts to a specific \((x, y)\) point to be plotted on a Cartesian plane.

Step 4: Adjust Angles for Negative Radii

When the radius is negative, adjust the angle by adding \(\pi\) to \(\theta\) in the polar coordinates:- For example, \((-2, \frac{1}{2}\pi)\) converts to \((2, \pi + \frac{1}{2}\pi)\). Apply this adjustment in conversion to get correct Cartesian coordinates.

Step 5: Repeated Angle Normalization

Normalize angles by adding or subtracting \(2\pi\) until they fall within the standard range \([0, 2\pi]\). This ensures angles like \(-7\pi\) or \(-\frac{33}{2}\pi\) are properly represented when calculating Cartesian points.

Step 6: Summary of Cartesian Plots

After conversion and normalization, plot the points:1. \((-2.12, 2.12)\)2. \((0, -2)\)3. \((1.73, 1)\)4. \((-1.54, -0.84)\)5. Repeat similar calculations for each point as detailed above.

Key Concepts

Cartesian coordinates

Cartesian coordinates represent points in a plane using two numbers:

• The x-coordinate (horizontal position)
• The y-coordinate (vertical position)

These concepts are visualized as a grid where each point corresponds to a unique (x, y) pair.
Cartesian coordinates make it easier to visualize geometrical concepts and mathematics.
Polar coordinates, on the other hand, use a radius and an angle to describe a point's position in a plane. Converting from polar to Cartesian is essential for plotting points on a standard graph.
The conversion involves using trigonometry functions. Specifically, using the angle (θ) and radius (r) with:

• \[x = r \cos(\theta)\]
• \[y = r \sin(\theta)\]

This transformation aligns polar descriptions into the familiar (x, y) plane.

angle normalization

Angles in polar coordinates can extend beyond the typical 0 to 2\(\pi\) radian range, which can complicate interpretations.
Angle normalization is the process of adjusting an angle to fit within this standard range. To normalize angles:

• If an angle exceeds 2\(\pi\), subtract 2\(\pi\) until it falls within the range.
• If an angle is negative, add 2\(\pi\) repeatedly.

For example, the angle \(-7\pi\) can be normalized:

• Add 2\(\pi\) three times to fit it into the 0 to 2\(\pi\) range.
• This ensures consistency in calculations and helps in converting to Cartesian coordinates accurately.

Normalization aligns angles, allowing easier manipulation and interpretation of the data.

negative radius adjustment

Unlike Cartesian coordinates, polar coordinates can have a negative radius. This points in the opposite direction given by the angle. Adjusting this involves reversing the direction by adding \(\pi\) to the angle:

• If the radius is negative, simply add \(\pi\) to the angle.
• For \((-2, \frac{1}{2}\pi)\), adjust to \(2, \pi + \frac{1}{2}\pi\)\ to convert its position properly.

This adjustment flips the point across the origin without changing its position relative to the original angle.
Handling negative radii correctly guarantees accurate plotting of points since it indicates direction rather than a backward step on the line.

coordinate conversion

Coordinate conversion bridges polar coordinates with the familiar Cartesian system, especially when plotting points.
Conversion capitalizes on trigonometric identities to map points effectively:

• Apply \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\)
• Perform angle normalization to fit within 0 to 2\(\pi\)
• Adjust negative radii by altering angles with \(\pi\)

These steps ensure that each point is properly translated to its Cartesian counterpart, easing graphing on a standard plane.
Accurate conversion ensures understanding of spatial relationships, advancing both educational insights and practical plotting of data.