Problem 13 - Thomas Calculus

Question

Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises \(7-22\) . $$ \theta=\pi / 3, \quad-1 \leq r \leq 3 $$

Step-by-Step Solution

Answer

Graph a line segment from \( r = -1 \) to \( r = 3 \) at \( \theta = \pi/3 \).

Step 1: Identify Polar Equation

The equation given is \ \( \theta = \frac{\pi}{3} \) \ which indicates all points on the graph will lie on a line through the origin making an angle of \( \frac{\pi}{3} \) radians with the positive x-axis.

Step 2: Determine Range for r

The inequality given is \ \( -1 \leq r \leq 3 \). \ This means that points will be plotted on the line \( \theta = \frac{\pi}{3} \) from \( r = -1 \) to \( r = 3 \). Note that \( r = -1 \) indicates a point on the opposite side of the origin.

Step 3: Graph Line Segment

To graph this, first locate the angle \( \frac{\pi}{3} \), which is equivalent to 60 degrees, on the polar coordinate plane. Draw a line from the origin extending in both directions at this angle. \Then, mark the segment of this line starting at \( r = -1 \), just opposite the origin, and ending at \( r = 3 \), away from the origin on the same line.

Key Concepts

graphing polar equations

Graphing polar equations can often seem challenging, but once you understand the coordination system and how the equations translate into graphical representations, it becomes much simpler. In this exercise, we're working with polar coordinates, which are used to determine a point on a plane using an angle and a radius (or distance). Unlike Cartesian coordinates, which use a grid of x and y values, polar coordinates use \(r\) for the radius and \(\theta\) for the angle made with the positive x-axis.

• The equation \(\theta = \frac{\pi}{3}\) indicates a specific direction or angle from the origin.
• This angle of \(\frac{\pi}{3} rad = 60\) degrees tells us the line along which our points will be situated.
• Points are plotted at various distances along this line, limited by the radius values allowed in the inequality given.

Understanding these components makes it easier to visualize the graph and plot the appropriate points.

coordinate system

The coordinate system is fundamental in plotting any graph, whether polar or Cartesian. In this case, we focus on the polar coordinate system:

• Each point is represented by a combination of \(r\) (radius) and \(\theta\) (angle).
• Unlike traditional Cartesian coordinates, points may have negative radii, indicating direction opposite to that which the angle suggests.
• This results in more flexibility, allowing for interesting paths such as circles, spirals, and the familiar line, as seen here.

Using polar coordinates requires us to think not only in terms of 'where' a point is but also 'how far' and 'at what angle' from the origin it is located. This adds a dimensional depth that Cartesian coordinates do not provide.

line segment in polar coordinates

The line segment in polar coordinates, seen here in this exercise, is essentially a series of points that align on a linear path stemming from an angle \(\theta\). A line segment's existence in polar coordinates is important for solving many equations:

• The given \(\theta = \frac{\pi}{3}\) defines the direction and \(-1 \leq r \leq 3\) defines the length or extent of the segment.
• This range from \(-1\) to \(3\) indicates a segment stretching from one side of the pole to the other, crossing the origin.
• Such a segment is visualized by plotting points starting from the direction opposite the initial \(r=1\) side to the ending point in the original direction.

This representation reminds us that in polar graphs, directionality is central, and understanding the length or segment of lines plotted augments our ability to accurately describe spatial designs and figures.