Problem 28 - A Graphical Approach to Precalculus with Limits

Question

Plot the point whose rectangular coondinates are given. Then determine nwo pairs of polar coondinates for the point with \(0^{\circ} \leq \theta<360^{\circ} .\) Do not use a calculator. $$(-2,0)$$

Step-by-Step Solution

Answer

The polar coordinates are (2, 180°) and (2, -180°).

Step 1: Understand the Rectangular Coordinates

The point given in rectangular coordinates is (-2, 0). This means the point is located 2 units to the left of the origin on the x-axis.

Step 2: Plot the Point

Draw a 2D coordinate system with x and y axes. From the origin (0,0), move 2 units to the left along the x-axis to place the point at (-2,0).

Step 3: Convert to Polar Coordinates

In polar coordinates, a point is represented by (r, \(\theta\)) where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle from the positive x-axis. For (-2,0), \(r=2\) since it is 2 units away from the origin along a line parallel to the x-axis. The angle \(\theta\) is 180 degrees since the point is on the negative side of the x-axis.

Step 4: Determine Two Pairs of Polar Coordinates

The first pair is (2, 180°). A second pair can use the same distance \(r=2\) but adjust the angle to \(\theta = -180\degree\), which is equivalent to 180° as angles are periodic with 360 degrees.

Step 5: Verify the Polar Coordinates

Check that both polar pairs (2, 180°) and (2, -180°) correctly terminate at the original rectangular coordinates (-2,0). Both represent the same location in the 2D plane.

Key Concepts

Rectangular Coordinates

Rectangular coordinates are a way to pinpoint the location of a point on a two-dimensional plane using two values: the x-coordinate and the y-coordinate. In most cases, this system is also referred to as the Cartesian coordinate system, named after the mathematician René Descartes.
In this system:

• The x-coordinate represents the horizontal distance from the vertical y-axis.
• The y-coordinate represents the vertical distance from the horizontal x-axis.

For the given problem, we have the coordinates ((-2, 0)). This tells us that the point is located exactly 2 units to the left of the origin (which is where the x and y axes intersect). The y-coordinate being 0 indicates that the point is neither above nor below the x-axis, but exactly on it.
When plotting points using rectangular coordinates, simply start at the origin and move horizontally to the left (or right) based on the x-coordinate and vertically up (or down) based on the y-coordinate.

Coordinate System

A coordinate system is a method used for describing the position of points or objects in a mathematical space. There are various types of coordinate systems, each designed for different types of problems and scenarios. Two of the most common are the rectangular (Cartesian) and polar coordinate systems.
The Cartesian coordinate system involves two perpendicular axes: the x-axis and the y-axis, which intersect at the origin (0,0). Each point is expressed in terms of these axes. However, for certain problems involving rotation or where a circular motion is more applicable, the polar coordinate system is often used.
In the polar coordinate system:

• The position of a point is represented by a radius, \(r\), and an angle, \(\theta\).
• \(r\) indicates how far the point is from the origin.
• \(\theta\) is the angle measured from the positive x-axis (also known as the polar axis).

This exercise involves converting a point from rectangular coordinates to polar coordinates, providing two different polar representations for the same point.

Angle Measurement

In the context of polar coordinates, the angle \(\theta\) is vital, as it determines the direction of the point from the origin. It's typically measured in degrees within the range of \(0^{\circ}\) to \(360^{\circ}\). Angles can also be measured in radians, but degrees are often used for simplicity, especially in introductory problems.

• When you measure an angle, you start from the positive x-axis and rotate counter-clockwise for positive angles.
• Negative angles are measured in the opposite, clockwise direction.

For the given problem, the point \((-2, 0)\) lies directly on the negative x-axis. This means the angle \(\theta\) needed to reach this point from the positive x-axis is \(180^{\circ}\).
Because of the periodic nature of angles, adding or subtracting multiples of \(360^{\circ}\) results in the same position on the circle. Thus, an angle of \(-180^{\circ}\) will also correctly describe the same point as \(180^{\circ}\). This ability to express a polar coordinate angle in multiple ways offers flexibility, which can be crucial in solving more complex problems.