Problem 17 - Algebra and Trigonometry

Question

Use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. $$ (-1, \pi) $$

Step-by-Step Solution

Answer

The point will be plotted one unit away along the positive x-axis.

Step 1: Understanding the Polar Coordinates

Recognize that the polar coordinate is given in the form (r, θ), where r is the distance from the origin and θ is the angle. Given polar coordinates are (-1, π). Here, r=-1 and θ=π.

Step 2: Plotting the Angle θ

Start with plotting the angle θ. The angle θ is π in the standard position, which lies along the negative x-axis if r had been positive.

Step 3: Consider the Negative r

Since r is negative, rather than extending the line from the origin along the direction of θ, we should extend the line in the opposite direction. Thus, take the line in the positive x-axis direction, since π is along the negative x-axis.

Step 4: Plotting the Dot

Plot the dot one unit away along the positive x-axis, because the magnitude of r is 1.

Key Concepts

Plotting Points in Polar Coordinates

In the polar coordinate system, each point on a plane is described by two values: the radius \( r \) and the angle \( \theta \). Unlike Cartesian coordinates, which use \( x \) and \( y \) to specify a location, polar coordinates rely on distance from a reference point (the origin) and direction.

• The radius \( r \) indicates how far the point is from the origin.
• The angle \( \theta \), measured in radians, shows the direction relative to the positive x-axis.

To plot a point like \((-1, \pi)\), start by understanding what these values denote. In this case, \( r = -1 \) and \( \theta = \pi \). Start by finding the direction for the angle \( \pi \), which is along the negative x-axis. After determining the direction, adjust for the radius. If the radius is negative, the actual point will be located in the opposite direction of the angle. This reversal of direction is important in fully understanding polar points.

Understanding Negative Radius

A negative radius in polar coordinates is a unique characteristic that can initially seem counterintuitive. Normally, the radius \( r \) indicates how far to move from the origin in the direction of the angle \( \theta \). However, when \( r \) is negative, it means that you need to move in the opposite direction. For example, with \((-1, \pi)\):

• The angle \( \pi \) positions the direction along the negative x-axis.
• Since \( r = -1 \), the point is plotted in the opposite direction along the positive x-axis because \(-r\) reverses the direction of \( \theta \).

Visualizing negative radii can be made easier by treating the value as a condition for changing direction. Always remember that the magnitude/absolute value remains the same; only the positioning direction changes. This concept highlights the flexibility and power of the polar coordinate system in representing positions on a plane.

Understanding the Polar Coordinate System

The polar coordinate system is a two-dimensional system where points are determined by a distance from a fixed central point and an angle from a fixed direction. This system is especially useful for circular and spiral patterns, as it describes the location more naturally than the Cartesian system for these forms.

• Origin: The center of the polar coordinate system is known as the pole (analogous to the origin in Cartesian coordinates).
• Angle \( \theta \): Measured in radians, counter-clockwise from the positive x-axis (similar to rotating around the origin).
• Radius \( r \): The distance from the pole.

The beauty of the polar coordinate system lies in its ability to simplify complex geometries like spirals or circles, where Cartesian equations might be cumbersome. By adapting these principles, plotting points even with conditions such as negative radii becomes intuitive. Understanding this system not only provides insight into unique coordinate challenges but also broadens comprehension in fields like physics and engineering.