Problem 8 - Calculus Early Transcendentals

Question

\(7-12\) Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. $$r \geqslant 0, \quad \pi / 3 \leqslant \theta \leqslant 2 \pi / 3$$

Step-by-Step Solution

Answer

The region is a sector extending from \( \theta = \pi/3 \) to \( \theta = 2\pi/3 \), with \( r \) starting at 0 and extending infinitely.

Step 1: Understanding Polar Coordinates

In polar coordinates, a point is defined by a distance from the origin, denoted as \( r \), and an angle \( \theta \) from the positive x-axis. The problem gives us conditions for both \( r \) and \( \theta \).

Step 2: Condition on r

The condition \( r \geq 0 \) specifies that the points are located on or outside the origin (positive radial direction). Since there is no upper limit, the points extend infinitely in the given directions specified by \( \theta \).

Step 3: Condition on θ

The angle \( \theta \) is restricted to \( \pi/3 \) to \( 2\pi/3 \). In a Cartesian system, this translates to angles from 60 degrees to 120 degrees. This creates a sector of a circle, starting from the origin.

Step 4: Sketching the Region

On the polar plane, sketch lines corresponding to \( \theta = \pi/3 \) and \( \theta = 2\pi/3 \), emerging from the origin. The region is the area between these two lines extending outward from the origin, infinitely, as there is no restriction on the distance \( r \).

Key Concepts

Radian Measure

Radian measure is a fundamental concept for working with angles in polar coordinates. It offers an alternative to degrees, providing a more natural and mathematically convenient way of expressing angles. A radian measures the angle created when the radius of a circle is wrapped along its circumference. This essentially forms a full circle using natural constants.
When converting between radians and degrees, remember that a full circle in degrees equals 360°, while in radians it's 2\( \pi \). Therefore, to convert degrees to radians, you multiply by \( \pi/180 \). Conversely, to get degrees from radians, multiply by \( 180/\pi \).
In our exercise, you're asked to work with angles between \( \pi/3 \) and \( 2\pi/3 \). Converting these to degrees helps visualize the sector: \( \pi/3 \) is 60°, and \( 2\pi/3 \) is 120°. Understanding this conversion is essential for interpreting and solving problems involving polar coordinates.

Coordinate Planes

The concept of coordinate planes plays a crucial role when discussing polar coordinates. Unlike the more familiar Cartesian coordinate system, which uses a grid of perpendicular lines to define points with an \(x\) and \(y\) axis, polar coordinates use a circle's radius and angle.
In the polar coordinate plane, the center is known as the pole (analogous to the origin in Cartesian coordinates). The angle \( \theta \) is measured from the positive \(x\)-axis, and the radius \( r \) represents the distance from the pole. Each point in this system is defined as \((r, \theta)\).
This radial system can be particularly helpful in situations involving circular or spherical objects, which simplifies equations and computations that might be complex using Cartesian coordinates. The visual representation on the polar coordinate plane helps in understanding and solving exercises, such as sketching regions based on angular sectors.

Angular Sectors

Understanding angular sectors is key to solving problems in polar coordinates. An angular sector is a portion of a circle, defined by two radii and the angle between them. This angle can be expressed in either radians or degrees, depending on the context.
For our exercise, where \( \pi/3 \leq \theta \leq 2\pi/3 \), you're identifying a sector of a circle centered at the origin. The sector stretches from 60° to 120°, covering the area between these two radial lines. Such regions can extend infinitely outwards because there's no upper limit on \( r \).
Visualizing this sector involves imagining a pizza slice: the crust represents the arc of the circle, and the two straight edges are formed by the radii at angles \( \pi/3 \) and \( 2\pi/3 \). This makes angular sectors particularly useful for defining and working with portions of circles in practical and theoretical contexts.