Problem 40 - Precalculus : Building Concepts and Connections

Question

For each of the points given in polar coordinates, find two additional pairs of polar coordinates \((r, \theta),\) one with \(r>0\) and one with \(r<0\). $$\left(\frac{7}{2}, 3 \pi\right)$$

Step-by-Step Solution

Answer

The two additional pairs of polar coordinates for \((\frac{7}{2}, 3 \pi)\), one with \(r>0\) and one with \(r<0\), are \((\frac{7}{2}, 3 \pi)\) and \((- \frac{7}{2}, 0)\), respectively.

Step 1: Interpret the Given Coordinate

The given polar coordinate is \((\frac{7}{2}, 3 \pi)\). Here, \(r=\frac{7}{2} > 0\) is the distance from the origin and \(\theta = 3 \pi\) is the angle from the positive x-axis.

Step 2: Find an Equivalent Representation with Positive r

This is trivial since the given coordinate already has \(r>0\). Hence, the first additional pair is \((\frac{7}{2}, 3 \pi)\).

Step 3: Find an Equivalent Representation with Negative r

For this, we need to find an equivalent point in the opposite direction (as a negative radius indicates that the point lies in the opposite direction), i.e., the angle \(\theta\) needs to be reversed by adding or subtracting \(\pi\) from it. If we add \(\pi\) to \(3 \pi\), it becomes \(4 \pi\) which is actually equivalent to \(0\) in the two-dimensional plane. Hence, we get the second additional pair as \((- \frac{7}{2}, 0)\).

Key Concepts

Polar Coordinate Representation

Polar coordinates provide an alternative way of representing points in a plane compared to the traditional Cartesian (or rectangular) coordinates. While Cartesian coordinates use a grid of horizontal and vertical lines to specify a point by its distance from a central reference point (the origin), polar coordinates do that by specifying the angle and distance of the point from a fixed direction and the origin, respectively.

The polar coordinate system uses two values: \( r \), the radial distance from the origin, and \( \theta \), the angle measured in radians or degrees from a fixed direction, usually the positive x-axis. Typically, \( \theta \) is measured counterclockwise, and when \( r > 0 \), the point is in the same direction as the angle \theta\. Conversely, if \( r < 0 \), the point is located in the opposite direction of \theta\.

Understanding polar coordinates is crucial in fields like physics and engineering where circular motion and periodic functions are often described naturally in terms of angles and radii.

Negative Radius in Polar Coordinates

The concept of a negative radius in polar coordinates can initially seem counterintuitive. In a rectangular coordinate system, distance is always positive, but in polar coordinates, the radius \( r \) can take on negative values.

A negative value of \( r \) implies that the point is located on the line that makes the angle \( \theta \), but in the opposite direction. Mathematically, you can think of this as the equivalent of adding or subtracting \( \pi \) radians or 180 degrees to the angle \( \theta \).

For example, the polar coordinate \((-r, \theta)\) is the same point as \((r, \theta + \pi)\) or \((r, \theta - \pi)\). This principle can be used to find equivalent polar coordinates with a negative radius, as illustrated by the exercise where the point \(\left(-\frac{7}{2}, 0\right)\) represents the same location as \(\left(\frac{7}{2}, 3\pi\right)\) on the Cartesian plane.

Equivalent Polar Coordinates

Equivalent polar coordinates are pairs or sets of \((r, \theta)\) values that identify the same point in the polar plane. This is possible because the polar system is periodic with respect to the angle and symmetric with respect to the radius.

For the angle \( \theta \), adding or subtracting multiples of \( 2\pi \) doesn't change the position of the point. Therefore, \( (r, \theta) \) and \( (r, \theta + 2k\pi) \), where \( k \) is an integer, represent the same point. In a similar fashion, since a negative radius flips the direction of the point 180 degrees, \( (r, \theta) \) and \( (-r, \theta + \pi) \) also describe the same point.

Using this understanding of equivalent polar coordinates, students can derive multiple representations of a single point. This is particularly beneficial when dealing with complex problems involving polar equations and graphing polar coordinates, as it provides flexibility in choosing the most suitable representation of the point.