Problem 36 - Precalculus : Building Concepts and Connections

Question

For each of the points given in polar coordinates, find two additional pairs of polar coordinates \((r, \theta),\) one with \(r>0\) and one with \(r<0\). $$(5,0)$$

Step-by-Step Solution

Answer

Two sets of polar coordinates for the point are \((5, 2\pi)\) with \(r>0\), and \((-5, \pi)\) with \(r<0\).

Step 1: Find polar coordinates with r>0

The exercise provides the coordinates (5,0). This one already has \( r>0 \). By adding \( 2\pi \) to the angle, we obtain another set of coordinates which represent the same point. Hence, the pair \((5, 2\pi)\) is another pair of polar coordinates for the point.

Step 2: Find polar coordinates with r<0

To find a pair of polar coordinates with \( r<0 \), we can 'flip' the point to the other side of the origin. We achieve this by adding \( \pi \) to the angle and making \( r \) negative. Hence, the pair \((-5, \pi)\) is another pair of polar coordinates for the point.

Key Concepts

Polar Coordinate System

The polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. This system is particularly useful for dealing with problems involving circular and spiral shapes, where traditional Cartesian (x, y) coordinates can be less intuitive.

In polar coordinates, the reference point is called the pole, often analogous to the origin in the Cartesian system, and the reference direction is usually the positive x-axis, from which angles are measured. A point's location is given as \(r, \theta\), where \(r\) is the radius - the distance from the pole, and \(\theta\) is the polar angle - the angle of rotation from the reference direction. Remember:

• The radius \(r\) can be any real number (positive, negative, or zero).
• The angle \(\theta\) is usually measured in radians, and while it can take any value, it is often expressed in the range \(0\) to \(2\pi\) for convenience.
• Due to the circular nature of angles, adding \(2\pi\) to the angle \(\theta\) does not change the point's location.

Understanding the polar coordinate system is crucial for solving many problems in mathematics, physics, and engineering, where symmetry around a point makes polar coordinates the sensible choice.

Negative Radius in Polar Coordinates

In the realm of polar coordinates, you might wonder what it means to have a negative radius, since we usually think of distance as a positive value. However, in this system, a negative radius is allowed and has a practical interpretation. When \(r\) is negative, it means that the point lies on the line that forms an angle \(\theta\) with the polar axis but extended in the opposite direction.

Here's how to visualize it:

• Imagine standing at the origin and facing the direction of \(\theta\).
• If \(r\) is positive, you'll walk forward in that direction by \(r\) units.
• If \(r\) is negative, you'll walk backward by \(r\) units, essentially moving in the opposite direction along the line of angle \(\theta\).

This concept allows for an additional set of polar coordinates to represent the same point. For example, if a point has polar coordinates \(5, 0\), it can also be represented by \(\-5, \pi\), meaning if you move 5 units in the direction opposite to angle 0 (or \(\pi\)), you end up at the same point.

Angle Addition in Polar Coordinates

Adding angles in polar coordinates is a critical operation. It allows us to find different polar representations of the same point, which is particularly useful in many applications like physics and engineering. When we add \(2\pi\) to the angle \(\theta\), the point completes a full circle around the pole and ends up at the same position.

To add angles effectively, keep these points in mind:

• With polar coordinates, angles are periodic, meaning that adding \(2\pi\) (a full rotation) to any angle \(\theta\) gives us an equivalent angle corresponding to the same direction.
• The set of all points that can represent the same location in a plane using polar coordinates is infinite because you can keep adding or subtracting \(2\pi\) to the angle.

For the example given in the exercise, adding \(2\pi\) to the angle of \(5, 0\) gives us the new coordinates \(5, 2\pi\), which describes the same point in the plane. This property is a fundamental aspect of the polar coordinate system, allowing for multiple representations of a single point.