Problem 13 - Calculus Early Transcendentals

Question

Graph the points with the following polar coordinates. Give two alternative representations of the points in polar coordinates. \(\left(-4, \frac{3 \pi}{2}\right)\)

Step-by-Step Solution

Answer

Answer: The two alternative representations of the point in polar coordinates are \(\left(-4, \frac{7\pi}{2}\right)\) and \(\left(-4, -\frac{\pi}{2}\right)\).

Step 1: Convert polar coordinates to Cartesian coordinates

To convert the polar coordinates \(\left(-4, \frac{3\pi}{2}\right)\) to Cartesian coordinates \((x, y)\), use the following equations: \(x = r\cos\theta\) \(y = r\sin\theta\) where \(r\) is the radial distance (magnitude) and \(\theta\) is the angle (direction) in radians. In this case, \(r = -4\) and \(\theta = \frac{3\pi}{2}\). \(x = -4 \cos \left(\frac{3\pi}{2}\right) = -4 \cdot 0 = 0\) \(y = -4 \sin \left(\frac{3\pi}{2}\right) = -4 \cdot (-1) = 4\) So, the Cartesian coordinates of the point are \((0, 4)\).

Step 2: Plot the point on a graph

Plot the point with the Cartesian coordinates \((0, 4)\) on a graph. This point is located on the positive y-axis, 4 units above the origin.

Step 3: Find alternative polar coordinates

To find alternative polar coordinates, add or subtract multiples of \(2\pi\) to the angle \(\theta\). For example: Alternative Representation 1: Add \(2\pi\) to \(\theta\) \(\theta' = \frac{3\pi}{2} + 2\pi = \frac{3\pi}{2} + \frac{4\pi}{2} = \frac{7\pi}{2}\) The point can also be represented as \(\left(-4, \frac{7\pi}{2}\right)\). Alternative Representation 2: Subtract \(2\pi\) from \(\theta\) \(\theta'' = \frac{3\pi}{2} - 2\pi = \frac{3\pi}{2} - \frac{4\pi}{2} = -\frac{\pi}{2}\) The point can also be represented as \(\left(-4, -\frac{\pi}{2}\right)\). Therefore, the polar coordinates \(\left[-4, \frac{3\pi}{2}\right]\) have two alternative representations: \(\left[-4, \frac{7\pi}{2}\right]\) and \(\left[-4, -\frac{\pi}{2}\right]\).

Key Concepts

Cartesian Coordinates

The Cartesian coordinate system is a way to determine the location of a point in a plane using a pair of values:

• The first value, often called the x-coordinate, indicates the point's horizontal distance from the origin (0,0).
• The second value, the y-coordinate, specifies the vertical distance from the origin.

This system is named after René Descartes, who introduced it. To convert from polar coordinates (which use a radius and angle) to Cartesian coordinates, we make use of trigonometric functions:

• Use the formula \(x = r \cos \theta\) to determine the x-coordinate.
• Use the formula \(y = r \sin \theta\) to calculate the y-coordinate.

In the problem we looked at, we converted the polar coordinates \((-4, \frac{3\pi}{2})\) to Cartesian coordinates. Given that \(r = -4\) and \(\theta = \frac{3\pi}{2}\), the calculations became:

• \(x = -4 \times 0 = 0\)
• \(y = -4 \times (-1) = 4\)

Thus, the Cartesian coordinates are \((0, 4)\), showing that the point is directly 4 units above the origin on the y-axis.

Graphing Points

Graphing points in the Cartesian plane involves placing points at specific locations based on their coordinates. For our derived point \((0, 4)\), we proceed as follows:

• Start at the origin—where the x-axis and y-axis intersect.
• Since the x-coordinate is 0, this means we need not move left or right.
• For the y-coordinate of 4, move up 4 units.

This places the point directly on the positive y-axis. Graphing is an essential skill because it visualizes data points and their relationships. Remember that the relative movement from the origin reflects the nature of the coordinates—with zero on one axis yielding straightforward plotting on the other axis.

Angle Conversion

Converting angles in polar coordinates is a significant part of understanding their flexibility. In polar coordinates, the angle \(\theta\) describes the direction from the positive x-axis. These angles are usually given in radians, and sometimes it’s helpful to find alternative expressions by adding or subtracting multiples of \(2\pi\).

• When we add \(2\pi\) (a full circle) to an angle, we effectively rotate the point around the circle back to its original position. For example, adding \(2\pi\) to \(\frac{3\pi}{2}\) gives us \(\frac{7\pi}{2}\).
• Similarly, subtracting \(2\pi\) leads us to another equivalent direction. For instance, \(\frac{3\pi}{2} - 2\pi\) results in \(-\frac{\pi}{2}\).

Converting angles like this helps to represent the same point using different polar coordinates, allowing for more flexibility in mathematical calculations and contextual understanding. In this exercise, the original point \((-4, \frac{3\pi}{2})\) could also be expressed as \((-4, \frac{7\pi}{2})\) or \((-4, -\frac{\pi}{2})\), demonstrating the cyclical nature of angular measurements in polar systems.