Problem 17 - Algebra and Trigonometry Real Mathematics, Real People

Question

Plot the point given in polar coordinates and find three additional polar representations of the point, using \(-2 \pi<\boldsymbol{\theta}<\mathbf{2} \pi\) $$(0,-\pi / 2)$$

Step-by-Step Solution

Answer

The point \((0,-\pi / 2)\) in three additional polar representations is \((0, \frac{3 \pi}{2})\), \((0, \frac{7 \pi}{2})\), \((0, \frac{11 \pi}{2})\). All these points are at the origin in the polar plane.

Step 1: Plot initial point

The given point is \((0,-\pi / 2)\). In the polar coordinate system, the first value denotes the radial coordinate (denoted as 'r' which is the distance from origin) and the second value is the angular coordinate (denoted as '\(\theta\)' which is the angle in radians). The point \((0,-\pi / 2)\) means the point is at origin, being 0 units away from the origin because the radial coordinate is zero. Therefore, no angular change occurs. Hence we start from the origin point.

Step 2: Find first additional representation

To find an additional representation, we can add \(2\pi\) (one complete revolution) to the angle. When we add \(2\pi\) to \(-\pi/2\), we get \(\theta = \(2\pi - \pi / 2\) = \(\frac{3\pi}{2}\). Since \(r=0\), the new point is \((0, \frac{3 \pi}{2})\). This point is still at origin as its radial coordinate is zero.

Step 3: Find second additional representation

Another additional representation could be found by subtracting \(2\pi\) from \(-\pi/2\). However, this will yield a result \(\theta = -\frac{5\pi}{2}\), which is less than \(-2 \pi\). Therefore, this result can not be used. Another approach could be adding or subtracting multiples of \(2\pi\) to/from the previously obtained valid angle, \(\frac{3 \pi}{2}\). This gives us \((0, \frac{3 \pi}{2} + 2\pi) = (0, \frac{7 \pi}{2})\) as the next valid point.

Step 4: Find third additional representation

Following similar procedure, adding another set of \(2\pi\) to \(\frac{7 \pi}{2}\) we get our next valid point, \((0, \frac{7 \pi}{2} + 2\pi) = (0, \frac{11 \pi}{2})\).

Key Concepts

Plotting Polar Coordinates

Understanding how to plot polar coordinates is an essential skill for visualizing complex numbers and solving problems in fields like physics and engineering. To begin plotting a point given in polar coordinates, we need to consider two components: the radial and angular coordinates. The radial coordinate represents the distance from the origin, while the angular coordinate indicates the direction from the positive x-axis.

For instance, a polar coordinate of \( (0, -\frac{\pi}{2}) \) signifies that the point lies at the origin because the radial distance is 0. Hence, no actual plotting away from the origin is necessary, and the angular direction becomes irrelevant. When working with polar coordinates, always start by locating the angle on a polar grid, then move outward from the origin by the radial distance. However, with a radial coordinate of 0, the point will remain at the center, regardless of the angle.

Polar Representations

Polar representations provide us with different ways of describing the same point in a polar coordinate system. This versatility is due to the periodic nature of angles, where adding or subtracting full rotations (\(2\pi\) radians) leads to equivalent positions. For example, the polar coordinate \( (0, -\frac{\pi}{2}) \) can represent a point at the origin.

Additional polar representations of this point can be found by altering the angular coordinate. As shown in the solution, by adding \(2\pi\) to the angle \( -\frac{\pi}{2} \), we obtain a new, but equivalent, angular coordinate of \( \frac{3\pi}{2} \) while keeping the radial coordinate at zero. Similarly, we can extend this process to ascertain further equivalent coordinates, showcasing the infinite number of ways to represent a single point in polar form.

Angular Coordinate

The angular coordinate in polar coordinates, denoted by \(\theta\), measures the counterclockwise angle from the positive x-axis to the point's projection on a circle centered around the origin. It is usually expressed in radians. The beauty of this coordinate is that it allows for infinite representations due to the circular nature of angles.

To add clarity to the concept, consider the initial angular coordinate of \( -\frac{\pi}{2} \) from the exercise. By simply adding \(2\pi\), we can alter the angle to find an equivalent representation, illustrating the periodic attribute of angular coordinates. When plotting, remember that positive angles sweep counterclockwise, while negative angles go clockwise. Keeping track of the angle's sign and magnitude is crucial in defining the point's precise location in the polar plane.

Radial Coordinate

The radial coordinate, often represented by 'r', measures how far a point lies from the pole or origin of the polar coordinate system. Its value can be any real number: positive, negative, or zero. A unique aspect of the radial coordinate is that, when it equals zero, the position of the point is exactly at the origin, rendering the angular coordinate inconsequential.

In the context of the exercise, the radial coordinate is zero, \( (0, -\frac{\pi}{2}) \), indicating a location at the polar origin. It's important to understand that negative values of 'r' can also be used, representing the point that lies on the line through the pole and the point corresponding to the positive value of 'r', but in the opposite direction. This feature of the radial coordinate allows for alternative expressions of the same point, thereby enriching the understanding of polar coordinate systems.