Problem 20 - 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. $$ (-3,-\pi) $$

Step-by-Step Solution

Answer

The point (-3,-π) in polar coordinates is plotted by moving to the π radians (or 180 degrees) counter-clockwise from the positive x-axis and then moving 3 units in the opposite direction because of the negative radius. It ends up on the positive x-axis, 3 units from the origin.

Step 1: Understand the significance of the negative radius

Under normal circumstances, a radius, being a distance, would not be negative. However, when reading polar coordinates, a negative radius simply means that we move in the opposite direction. So, for a point with radius -3, instead of moving 3 units out in the direction of the angle given, we will be moving 3 units in the exact opposite direction.

Step 2: Convert the negative radian measure to a positive angle

The angle -π is the same as an angle of π radians, so this point is located at an angle of π radians from the positive x-axis. This is equivalent to 180°.

Step 3: Plot the point

The point is plotted by first determining the direction. From the origin, pivot π radian along the positive x-axis which is equivalent to moving along to the leftmost point of the coordinate circle. Then, instead of moving 3 units out in the direction of π, move 3 units towards the right to the exact opposite direction because of the negative radius, and that is your point.

Key Concepts

Understanding Negative Radius in Polar Coordinates

In the realm of polar coordinates, understanding the concept of a negative radius can be quite intriguing. Typically, the radius in a coordinate system is a measure of distance and is positive. However, when dealing with polar coordinates, a negative radius means moving in the opposite direction from what the angle suggests. For example, if a coordinate is given as \((-3, -\pi)\), rather than moving 3 units away from the origin towards the angle \(-\pi\), you shift 3 units in the opposite direction.

This principle of direction reversal serves as a crucial part of plotting points in polar coordinates and helps you grasp unusual coordinate forms. It ensures precision in locating points on a plane, especially when working with various coordinates that include negative values.

Angle Conversion: Positive and Negative Angles

In polar coordinates, angles are vital in determining a point's direction. Angles can be expressed in radians or degrees and can be both positive and negative. The task is often to convert a negative angle into a more understandable, positive angle. As in our example, \(-\pi\) radians is equivalent to \(\pi\) radians. In degree measure, this is 180°. Knowing this conversion helps you easily visualize the direction.

To convert a negative angle to its positive counterpart, you can add \(2\pi\) radians (or 360°) to the negative angle. This maneuver keeps the angle within a standard range typically between 0 and \(2\pi\) radians. Such conversions make directional plotting straightforward and eliminate confusion in reading polar coordinates.

Plotting Polar Points with Negative Radius

Plotting polar points, especially those with a negative radius, involves a unique approach. Begin with identifying the angle, as in our \((-3, -\pi)\) example. The angle \(-\pi\) gets converted to \(\pi\) or 180°, meaning the direction lies on the negative x-axis.

Next, consider the radius. A positive radius would mean moving along the referenced angle; however, a negative radius implies moving in the exact opposite direction. Hence, with a radius of -3, you move 3 units from the origin towards the positive x-axis, even though the angle itself suggests leftward or negative x-direction.

This approach requires an understanding of both angle and radius adjustments. It's essential to ensure accuracy in plotting, utilizing these steps effectively to navigate the polar coordinate system efficiently.