Problem 11 - Contemporary Precalculus

Question

List four other pairs of polar coordinates for the given point, each with a different combination of signs (that is, \(r > 0, \theta > 0 ; r > 0, \theta < 0 ; r < 0, \theta > 0 ; r < 0, \theta < 0)\). $$(\sqrt{3}, 3 \pi / 4)$$

Step-by-Step Solution

Answer

Question: List four other pairs of polar coordinates for the given point \((\sqrt{3}, 3\pi/4)\). Answer: The four other pairs of polar coordinates for the given point are: 1. \((\sqrt{3}, 3\pi/4)\) 2. \((\sqrt{3}, -5\pi/4)\) 3. \((-\sqrt{3}, \pi/4)\) 4. \((-\sqrt{3}, -7\pi/4)\)

Step 1: Original Point:

The original point is given as \((\sqrt{3}, 3\pi/4)\) which is already in the form of \(r>0\), \(\theta>0\).

Step 2: Pair 1: \(r>0\), \(\theta<0\)

To find the coordinates with \(r>0\) and \(\theta<0\), we don't change the \(r\) value but need to make \(\theta\) negative. Since the period of \(\theta\) is \(2\pi\), we can subtract \(2\pi\) from the original \(\theta\) value. So, the new pair of polar coordinates will be \((\sqrt{3}, 3\pi/4 - 2\pi) = (\sqrt{3}, -5\pi/4)\).

Step 3: Pair 2: \(r 0\)

For the coordinates with \(r<0\) and \(\theta>0\), we need to change the sign of \(r\) and keep the \(\theta\) positive. Since the given point is in the 2nd quadrant, we have to subtract \(\pi\) from the original \(\theta\) to make the new point lie in the 4th quadrant. The new polar coordinates will be \((-\sqrt{3}, 3\pi/4 - \pi) = (-\sqrt{3}, \pi/4)\).

Step 4: Pair 3: \(r<0\), \(\theta<0\)

Finally, to obtain the coordinates with \(r<0\) and \(\theta<0\), both \(r\) and \(\theta\) should be negative. We already converted \(r\) to a negative value in the previous step. Now to make the \(\theta\) value negative, we subtract \(2\pi\) from the positive angle found in Pair 2, which was \(\pi/4\). So, the new polar coordinates will be \((-\sqrt{3}, \pi/4 - 2\pi) = (-\sqrt{3}, -7\pi/4)\). In conclusion, the four other pairs of polar coordinates for the given point are as follows: 1. \((\sqrt{3}, 3\pi/4)\) 2. \((\sqrt{3}, -5\pi/4)\) 3. \((-\sqrt{3}, \pi/4)\) 4. \((-\sqrt{3}, -7\pi/4)\)

Key Concepts

Polar Coordinate System

The polar coordinate system offers an alternative to the Cartesian coordinate system for representing points in a plane. Unlike the Cartesian system, which uses horizontal and vertical distances to define a location, the polar coordinate system measures the direction and distance of a point from a fixed central point, known as the pole (similar to the origin in Cartesian coordinates).

In this system, any point can be identified by two values: the radial coordinate, denoted as r, which is the distance from the pole, and the angular coordinate, denoted as \(\theta\), which is the angle measured from a fixed direction, usually the positive x-axis. Positive values of r indicate that the point is away from the pole, while negative values imply that the point is in the opposite direction after crossing the pole. Angles are usually measured in radians, where a complete rotation around the pole equals 2\pi radians.

Converting Polar Coordinates

Converting between different polar coordinates for a given point allows for the exploration of symmetry and periodic functions in polar equations. As seen in the original exercise, a single point can be represented by various combinations of r and \(\theta\).

To perform such conversions, it's important to understand that the angular coordinate \(\theta\) can be adjusted by adding or subtracting multiples of 2\pi, which corresponds to full rotations around the polar axis. Changing the sign of the radial coordinate, r, inverses the direction, effectively rotating the point by \pi radians. This allows for a multitude of representations, all valid within the system, greatly enhancing the flexibility and utility of the polar coordinates in solving complex problems and analyzing circular motion or rotation symmetry.

Angular Coordinate \(\theta\)

The angular coordinate, \(\theta\), represents the angle of rotation required to reach the point from the polar axis, which is conveniently chosen as the positive x-axis in a Cartesian system. It is one of the defining parameters in the polar coordinate system, and is measured in radians or degrees.

For describing locations precisely, the direction in which \(\theta\) is measured matters. In mathematics, angles in polar coordinates are typically measured in a counter-clockwise direction from the polar axis. It is worth noting that if an angle is negative or greater than 2\pi, it simply means the angle has been rotated beyond a full circle, and corresponds to a standard position when reduced to an equivalent angle between 0 and 2\pi. Understanding the concept of angular coordinates is essential for students tackling polar equations and graphs, as it influences the plot location of points in polar space.

Radial Coordinate r

The radial coordinate, denoted as r, indicates the distance from the pole, or origin, to the point in question. Positive values of r mean the point is located directly along the angle \(\theta\) from the pole. Negative values of r are used to indicate that the point is positioned on the line that extends from the pole through \(\theta\), but it is on the opposite side of the pole.

This coordinate is a key feature of the polar coordinate system, as it allows for circular and spiral shapes to be expressed and analyzed in a straightforward manner. When students work on problems involving the radial coordinate, they must pay attention to its sign and magnitude, as both influence the final position of the point in the polar plane. Through the exploration of radial coordinates, students gain an understanding of how distances and directions work in tandem to define the location of points in two-dimensional space.