Problem 15 - Thomas Calculus

Question

Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises \(11-26 .\) $$0 \leq \theta \leq \pi / 6, \quad r \geq 0$$

Step-by-Step Solution

Answer

Shade the sector from \( \theta = 0 \) to \( \theta = \pi/6 \) in the polar plane.

Step 1: Identify the limits of θ

We are given the inequality \( 0 \leq \theta \leq \frac{\pi}{6} \). This means that the angle \( \theta \) starts from \( 0 \) radians and goes up to \( \frac{\pi}{6} \) radians. In degrees, this corresponds to angles between \( 0^\circ \) and \( 30^\circ \).

Step 2: Determine the range for r

The inequality \( r \geq 0 \) indicates that we are interested in all points where the radial distance \( r \) is non-negative. This allows for points originating at the pole and extending outward indefinitely within the specified angular range.

Step 3: Analyze the given constraints in a polar graph

On a polar graph, represent the range for \( \theta \) by drawing two lines: one at \( \theta = 0 \) and the other at \( \theta = \frac{\pi}{6} \). Since \( r \) can be any non-negative value, shade the region starting from the origin and extending outward within these two lines.

Step 4: Graph the solution area

The solution consists of all points within the shaded region that lies between the lines \( \theta = 0 \) and \( \theta = \frac{\pi}{6} \) and extends from the origin along lines of increasing \( r \). The shaded sector of the circle represents the set of points that satisfy both the angle and radius conditions. This sector slices out a piece of the circle starting from the positive x-axis (\( \theta = 0\)) up to \( 30^\circ\).

Key Concepts

Angle Measurement in Polar Coordinates

In polar coordinates, the angle \( \theta \) plays a crucial role in determining the direction of a point from the origin. Unlike the Cartesian coordinate system, which uses (x, y) to locate a point, polar coordinates use \( (r, \theta) \), where \( \theta \) indicates the rotation from the positive x-axis.

• \( \theta \) is usually measured in radians, but it can also be converted to degrees. For example, \( \frac{\pi}{6} \) radians is equivalent to \( 30^\circ \).
• The range of angles represents the entire direction in which the point can move, originating from the pole (origin).
• For the equation \( 0 \leq \theta \leq \frac{\pi}{6} \), the angle is limited to a small sector between the positive x-axis and 30 degrees.

Understanding the angle is fundamental as it sets the boundaries for where any given point may lie on a polar graph.

Understanding Radial Distance

The radial distance \( r \) in polar coordinates measures how far away a point is from the origin, or pole. It serves as the second critical component alongside the angle \( \theta \).

• \( r \) can take any non-negative value, indicating that a point can be infinitely far from the origin within the given angle sector.
• The principle `\( r \geq 0 \)` tells us that points cannot be located behind the origin along the radial line.
• This distance is crucial for graphing as it dictates how extended the graph will appear from the pole outward.

In this context, all points starting from the pole and moving outwards will be included within the specified angle range. The flexibility of \( r \) allows the sector to stretch infinitely, presenting the complete picture of possibilities on a polar graph.

Polar Graphing Techniques

Visualizing data in polar form can sometimes be more intuitive – akin to creating a slice of pie on a circular graph. With specified parameters like those given, we can effectively map areas on the graph.

• Start by plotting the direction lines for \( \theta = 0 \) and \( \theta = \frac{\pi}{6} \), which helps encapsulate the angular range.
• Since \( r \geq 0 \), fill the area that starts from the origin outward to form a wedge-like region.
• This section of the polar plane is a crucial visualization tool for understanding the spatial boundaries set by the polar equation constraints.

By shading the area between these two lines extending from the origin, one visualizes the sector that satisfies all point conditions. Polar graphing provides a dynamic way to grasp the concept of radial and angular limits in action, portraying both the reach of radial distances and their angular limitations.