Problem 61 - Calculus Early Transcendentals

Question

Determine whether the following statements are true and give an explanation or counterexample. a. The point with Cartesian coordinates (-2,2) has polar coordinates \((2 \sqrt{2}, 3 \pi / 4),(2 \sqrt{2}, 11 \pi / 4),(2 \sqrt{2},-5 \pi / 4),\) and \((-2 \sqrt{2},-\pi / 4)\) b. The graphs of \(r \cos \theta=4\) and \(r \sin \theta=-2\) intersect exactly once. c. The graphs of \(r=2\) and \(\theta=\pi / 4\) intersect exactly once. d. The point \((3, \pi / 2)\) lies on the graph of \(r=3 \cos 2 \theta\). e. The graphs of \(r=2 \sec \theta\) and \(r=3 \csc \theta\) are lines.

Step-by-Step Solution

Answer

Answer: Statements a, b, and c are true, while statements d and e are false.

Step 1: a. Converting Polar Coordinates to Cartesian

First, verify if the given polar coordinates correspond to the Cartesian coordinates (-2, 2). The formula for conversion is: x = r * cos(θ), y = r * sin(θ). Convert the polar coordinates and check if the results match (-2, 2). 1. (2 * sqrt(2), 3 * pi / 4): x = 2 * sqrt(2) * cos(3 * pi / 4) = -2 y = 2 * sqrt(2) * sin(3 * pi / 4) = 2 2. (2 * sqrt(2), 11 * pi / 4): x = 2 * sqrt(2) * cos(11 * pi / 4) = -2 y = 2 * sqrt(2) * sin(11 * pi / 4) = 2 3. (2 * sqrt(2), -5 * pi / 4): x = 2 * sqrt(2) * cos(-5 * pi / 4) = -2 y = 2 * sqrt(2) * sin(-5 * pi / 4) = 2 4. (-2 * sqrt(2), - pi / 4): x = -2 * sqrt(2) * cos(- pi / 4) = -2 y = -2 * sqrt(2) * sin(- pi / 4) = 2 All the given polar coordinates correspond to the Cartesian coordinates (-2, 2). The statement is true.

Step 2: b. Intersection of graphs

First, we'll rewrite the equations r*cos(θ) = 4 and r*sin(θ) = -2 as x = 4 and y = -2, then see if they intersect exactly once. The graphs of x = 4 and y = -2 are both lines, and they intersect only once. The statement is true.

Step 3: c. Intersection of r=2 and θ=π/4

The graph of r=2 is a circle with radius 2, centered at the origin. The graph of θ=π/4 is a line passing through the origin, making a 45° angle with the x-axis. These two graphs only intersect at one point, hence, the statement is true.

Step 4: d. Verifying the point on the graph

We have the point (r, θ) = (3, π/2). We'll check if it satisfies the equation r = 3 * cos(2 * θ). r = 3 * cos(2 * (π/2)) = 3 * cos(π) = -3. This point does not satisfy the equation since r is not equal to -3. The statement is false.

Step 5: e. Identifying line graphs

The graphs of r=2*sec(θ) and r=3*csc(θ) are not lines. The graph of r=2*sec(θ) is the vertical polar asymptote. Similarly, r=3*csc(θ) is the horizontal polar asymptote. Neither of these are lines. The statement is false. Here are the results for each statement: a. True, b. True, c. True, d. False, e. False.

Key Concepts

Cartesian to Polar Conversion

Understanding the relationship between Cartesian and polar coordinates is a fundamental concept in calculus and analytical geometry. Cartesian coordinates represent points on a plane using x (horizontal) and y (vertical) values, whereas polar coordinates represent points by their distance from the origin (r) and the angle (θ) they make with the positive x-axis.

Converting from Cartesian to polar coordinates involves using the equations:

• For the radius: \( r = \sqrt{x^2 + y^2} \)
• For the angle: \( \theta = \arctan(\frac{y}{x}) \) for x > 0 or \( \theta = \arctan(\frac{y}{x}) + \pi \) for x < 0

The conversion process allows us to express a point that lies in a plane in another way, which can be more convenient depending on the context. For example, some curves are more naturally described in polar coordinates, such as circles and spirals.

As in the provided exercise where the Cartesian coordinates (-2, 2) can be converted into various sets of polar coordinates, it's important to note that unlike Cartesian coordinates, polar coordinates are not unique. Adding or subtracting multiples of \(2\pi\) to the angle gives rise to the same point in Cartesian coordinates. Hence, a single Cartesian point can be represented by infinite polar coordinates by changing the angle by full rotations.

Intersection of Graphs

The intersection of graphs represents the set of points that are common to all graphs being considered. In polar and Cartesian systems, understanding where curves intersect with each other can be crucial for solving problems involving areas, lengths, and other properties of shapes.

When dealing with the intersection of graphs, it is often useful to rewrite polar equations in their Cartesian form to better visualize and determine the points of intersection. For instance, the equation \(r \cos \theta = 4\) can be recognized as a vertical line (x=4) in the Cartesian system, and \(r \sin \theta = -2\) corresponds to a horizontal line (y=-2). These two lines will intersect at exactly one point, which can be represented as (4, -2) on the Cartesian plane.

In the exercise, it's confirmed that the graphs represented by the polar equations do intersect once, proving the statement true. It underscores the eternal dance between geometry and algebra; while the algebraic representation gives us a quick answer, being able to interpret it geometrically offers a more profound understanding.

Polar Graph Properties

Polar graphs have unique properties that differentiate them from their Cartesian counterparts. One key property is their symmetry. Polar equations can exhibit different types of symmetry such as symmetry about the polar axis (x-axis in Cartesian coordinates), symmetry about the pole (origin), or symmetry about the line \(\theta = \frac{\pi}{2}\).

Another important property is that polar graphs can represent many distinct points with the same radius but different angles, as multiple angles can correspond to the same point on the Cartesian plane, which we saw in the conversion exercise where the Cartesian point (-2, 2) corresponded to various polar coordinates.

Additionally, certain polar equations like \(r = 2\) represent simple shapes – in this case, a circle with radius 2. The equation \(\theta = \frac{\pi}{4}\), on the other hand, represents a straight line that makes a 45° angle with the horizontal axis. This highlights another powerful aspect of polar coordinates: the ability to represent complex curves and shapes in a simplified form, which can be invaluable in various branches of mathematics and physics. In the problem given, correctly understanding these properties helps in verifying that the graphs of \(r=2\) and \(\theta=\frac{\pi}{4}\) intersect exactly once.