Problem 18 - Thomas Calculus

Question

Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises \(7-22\) . $$ 0 \leq \theta \leq \pi, \quad r=-1 $$

Step-by-Step Solution

Answer

Graph a line segment from (-1, 0) to (1, 0) on the negative x-axis.

Step 1: Understand the Polar Coordinates

Polar coordinates represent a point in a plane with two values: \(r\) for radius (distance from the origin) and \(\theta\) for angle (measured from the positive x-axis). In this problem, \(r = -1\) and \(0 \leq \theta \leq \pi\).

Step 2: Interpret the Negative Radius

A negative radius in polar coordinates means the point is located in the direction directly opposite to that indicated by the angle \(\theta\). It can be interpreted as traveling 1 unit in the negative direction from the angle \(\theta\).

Step 3: Determine the Set of Points

For each angle \(\theta\) in the range from 0 to \(\pi\), the corresponding point is at a distance of 1 unit from the origin, in the direction opposite to \(\theta\). This traces a line segment where \(r = -1\).

Step 4: Graph the Solution

To graph these points, draw a line segment on the negative x-axis, starting at (-1, 0) and ending at (1, 0). This segment corresponds to \( r = -1 \) for the specified range of \( \theta \), since the negative radius moves the segment leftward.

Key Concepts

Negative Radius Interpretation

In polar coordinates, a negative radius may initially seem confusing, but it's quite simple once you understand the mechanics. Normally, \(r\) represents the distance from the origin. However, when \(r\) is negative, the point appears in the exact opposite direction of the angle \(\theta\). This means instead of moving outward from the origin in the direction \(\theta\), you move the same distance inward, creating a reflection across the origin.
To visualize, when given \(r = -1\), instead of moving towards the point on your graph with angle \(\theta\), you place the point one unit away on the opposite side. This is crucial when sketching because every angle \(\theta\) from 0 to \(\pi\) will map to the line segment from \(-1, 0\) to \(1, 0\), tracing back towards the origin. Always remember:

• Negative \(r\) means reflection through the origin.
• Points are plotted opposite to \(\theta\).

Polar to Cartesian Conversion

When graphed in Cartesian coordinates, polar coordinates need translating into familiar x and y coordinates. The conversion uses these relationships:

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

For our exercise where \(r = -1\) and \(0 \leq \theta \leq \pi\), let's convert the extremes. At \(\theta = 0\), using the cosine function because sine equals zero at this angle, we calculate:

• \[x = -1 \times \cos(0) = -1\]
• \[y = -1 \times \sin(0) = 0\]

This point is \((-1, 0)\). At \(\theta = \pi\), calculations show:

• \[x = -1 \times \cos(\pi) = 1\]
• \[y = -1 \times \sin(\pi) = 0\]

This becomes \((1, 0)\). Seeing these conversions helps to verify graph outputs.

Graphing Techniques in Polar Coordinates

Graphing polar coordinates can seem novel, but knowing specific techniques simplifies the process. One such example is maintaining awareness of an \(r\) value's sign—as seen with negative radius—since it dictates direction opposite to the angle.
For our specific problem, with a fixed \(r = -1\) and \(0 \leq \theta \leq \pi\), the goal is plotting:

• Start at the positive-direction angle, then reflect.
• Connect points across the range of \(\theta\) to form a line segment.

This method ensures accuracy as you graph from \((-1, 0)\) to \(1, 0)\), effectively plotting by consistent reflection across the angle \(\theta\).
These considerations reduce error and foster an intuitive grasp of plotting from polar instructions, offering further advantage with practice and visualization.