Problem 35 - 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 \(r<0\). $$\left(4, \frac{3 \pi}{2}\right)$$

Step-by-Step Solution

Answer

The additional pairs of polar coordinates for \( (4, \frac{3 \pi}{2})\) are \( (4, \frac{7 \pi}{2})\) and \( (-4, \frac{5 \pi}{2})\).

Step 1: Find equivalent positive radius representation

Given a point \( (4, \frac{3 \pi}{2})\) in polar coordinates, an equivalent point with same positive radius 'r' can be found by adding \(2\pi\) to \(\theta\). So, \(\theta' = \theta + 2\pi = \frac{3 \pi}{2} + 2\pi = \frac{7 \pi}{2}\). Therefore, one of the polar representation with \(r>0\) is \( (4, \frac{7 \pi}{2})\).

Step 2: Find equivalent negative radius representation

For finding a representation with a negative radius, we can find it by negating 'r' and adding \(\pi\) to \(\theta\). As such, \(r'=-r=-4\) and \(\theta' = \theta + \pi = \frac{3 \pi}{2} + \pi = \frac{5 \pi}{2}\). Consequently, an additional polar coordinate representation with \(r<0\) is \(-4, \frac{5 \pi}{2}\).

Key Concepts

Radius Conversion

In polar coordinates, radius conversion helps us find different ways to represent the same point. The radius is represented by 'r'. By changing the radius from positive to negative or vice versa, you can still describe the same location.
If you start with a point that has a positive radius, like \( (4, \frac{3\pi}{2}) \), you can transform this point by using angle modifications to find an equivalent with a negative radius. This simply involves changing the 'r' value to negative. In our example it becomes \( -4 \).
Switching the radius’ sign is an easy way to explore negative and positive equivalents while working in polar coordinates. These adjustments offer more flexibility and understanding when dealing with points in this system.

Angle Addition

Angle addition is another crucial concept when dealing with polar coordinates. It allows us to transform the angle component of a polar coordinate, known as \( \theta \).
By adding an angle (often \( 2\pi \) or \( \pi \)), you can find new equivalents for a given point. For example:

• Add \( 2\pi \) to a given angle to find another positive radius representation.
• In our exercise, to change the angle from \( \frac{3\pi}{2} \) to \( \frac{7\pi}{2} \), we added \( 2\pi \).
• To convert the radius to a negative, add \( \pi \) to the angle, turning \( \frac{3\pi}{2} \) into \( \frac{5\pi}{2} \).

These operations don't actually change the point's location, which remains the same on the polar plane. It means that angle addition is a method to reconfigure the representation for better clarity or specific application needs.

Polar Representation

Polar representation involves using a coordinate system that focuses on a point's distance from a central origin (radius) and the angle from a reference direction (angle, \( \theta \)). It's a unique way to present locations leverageable in both mathematical and real-world contexts.
Each polar coordinate is expressed as \( (r, \theta) \). In our exercise given with \( (4, \frac{3\pi}{2}) \), this means the point is 4 units from the origin, aligned at an angle of \( \frac{3\pi}{2} \) from the x-axis in standard position.
This system is especially beneficial when dealing with circular and periodic functions, allowing for easier descriptions and calculations than Cartesian coordinates might provide. Embracing both positive and negative radius conversions, along with angle additions, extends the utility and flexibility of polar representation to explore geometric interpretations seamlessly.