Problem 18 - Calculus

Question

True or false, with a reason or an example: (a) Changing to \(-r\) and \(-\theta\) produces the same point. (b) Each point has only one \(r\) and \(\theta,\) when \(r<0\) is not allowed. (c) The graph of \(r=1 / \sin \theta\) is a straight line.

Step-by-Step Solution

Answer

(a) False; (b) False; (c) True.

Step 1: Evaluate statement (a)

Statement (a) says that changing to \(-r\) and \(-\theta\) produces the same point. In polar coordinates, the point \((r, \theta)\) is the same as \((-r, \theta + \pi)\). Changing \(r\) to \(-r\) keeps the point in the opposite direction. If \(\theta\) is also changed to \(-\theta\), this effectively rotates the point in the opposite quadrant. Hence, this statement is **false** as there will be a different location.

Step 2: Evaluate statement (b)

Statement (b) states that each point has only one \(r\) and \(\theta\) with \(r < 0\) not allowed. In polar coordinates, multiple representations for a point exist, such as \((r, \theta + 2\pi n)\) or \((-r, \theta + \pi)\) for integer \(n\). Therefore, this statement is **false** because points can have different sets of \(r\) and \(\theta\) when \(r\) is positive.

Step 3: Evaluate statement (c)

Statement (c) investigates if the graph of \(r = \frac{1}{\sin \theta}\) is a straight line. Rewriting \(r = \frac{1}{\sin \theta}\) in Cartesian coordinates gives \(r = \frac{1}{y/r}\), simplifying to \(y = 1\). This equation indeed represents a horizontal line, not parallel to the polar axis. Thus, the statement is **true**.

Key Concepts

Coordinate Transformation

In the realm of polar coordinates, transforming one form into another can often unveil new perspectives about a location. Polar coordinates involve two main variables: the radius \(r\) and the angle \(\theta\). Unlike Cartesian coordinates, where every point has a unique \((x, y)\) pair, polar coordinates can describe the same point in multiple ways.
When we transform the coordinates by altering \(r\) to \(-r\) and \(\theta\) to \(-\theta\), it's crucial to understand how these transformations affect the point's position. The original point \((r, \theta)\) translates to \((-r, \theta + \pi)\) when only \(r\) changes, keeping the point in the opposite direction on the same radial line.
If we also change the angle to \(-\theta\), we further rotate the position across quadrants, leading to a distinct point. This is why altering both \(r\) and \(\theta\) typically results in a different point.
To visualize transformations effectively, think about the compass: adjusting \(r\) flips the direction, while \(\theta\) shifts the needle across the dials.

Polar Equations

Polar equations provide another fascinating way to describe curves and lines. They utilize \(r\) and \(\theta\) to define relationships, which can often appear quite elegant in comparison to their Cartesian counterparts. Here, let's take a closer look at the polar equation \(r = \frac{1}{\sin \theta}\).
At first glance, it might not be immediately clear what kind of shape this equation describes. Converting from polar to Cartesian form, we realize that \(r\) in terms of \(y\) (with \(r = y/\sin \theta\)) simplifies to \(y = 1\). This indicates that, in the Cartesian plane, the graph is a straight horizontal line at \(y = 1\).
It's crucial to remember that in polar coordinates, equations can yield results that differ in appearance and interpretation once moved to Cartesian coordinates. This transformation often helps in grasping the true nature of the given equation.

Graph Interpretation

Understanding graphs in different coordinate systems can open up a new dimension of interpretation. For polar coordinates, graph interpretation involves looking at how \(r\) variations with \(\theta\) manifest visually.
For example, consider the statement analyzed previously: "The graph of \(r = \frac{1}{\sin \theta}\) is a straight line." When plotted in polar coordinates, the line described doesn't look like the straight lines we're accustomed to in Cartesian graphs. However, when we convert it, we observe that it translates to \(y = 1\).
In the Cartesian system, this is easy to understand as a simple horizontal line. Yet, in polar coordinates, the concept of straight changes based on how radial distances and angles pair to form paths on the plane. Hence, always consider converting between coordinate systems to better comprehend the behaviors and shapes involved.