Problem 9 - The Calculus with Analytic Geometry

Question

Plot the point having the given set of polar coordinates; then give two other sets of polar coordinates of the same point, one with the same value of \(r\) and one with an \(r\) having opposite sign.\(\left(-4, \frac{5}{6} \pi\right)\)

Step-by-Step Solution

Answer

(4, \frac{11}{6} \pi) and (4, \frac{5}{6} \pi)

Step 1 - Understand Polar Coordinates

Polar coordinates \((r, \theta)\) are defined by the distance from the origin, \(r\), and the angle from the positive x-axis, \(\theta\). Here, \((r, \theta) = (-4, \frac{5}{6} \pi)\). The negative \(-4\) means the point is in the opposite direction of \(\theta\).

Step 2 - Plot the Point

To plot \((-4, \frac{5}{6} \pi)\), locate the angle \(\frac{5}{6} \pi\) (which is 150 degrees) and move 4 units in the opposite direction, since \(r\) is negative.

Step 3 - Convert to Positive Radius

To find coordinates with the same radius \(-4\) but with a positive radius, add \pi\ (180 degrees) to the angle: \(\frac{5}{6} \pi + \pi = \frac{5}{6} \pi + \frac{6}{6} \pi = \frac{11}{6} \pi\). Therefore, the coordinates are \(4, \frac{11}{6} \pi\).

Step 4 - Find Another Coordinate with Opposite Sign

To find another set with \(r\) having an opposite sign, maintain the angle \(\frac{5}{6} \pi\) and simply take the positive value of \(r\): \(4, \frac{5}{6} \pi\).

Key Concepts

Negative Radius

In polar coordinates, the radius \(r\) can be either positive or negative. A positive radius points directly in the direction of the angle \( \theta \), while a negative radius points exactly opposite.
For example, the point \((-4, \frac{5}{6} \pi)\) has a radius of -4. This means it lies exactly opposite to the angle \(\frac{5}{6} \pi\).

To plot this point, start by locating the angle \(\frac{5}{6} \pi\), which is 150 degrees. Instead of moving 4 units in this direction, move 4 units in the directly opposite direction.


Negative radius can be confusing at first, but just remember it inverts the position of the point through the origin.

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Angle Conversion

Angle conversion is a key concept when working with polar coordinates. There are often multiple ways to represent the same point using different angles.

To convert an angle to a positive radius, add \(\pi\) (180 degrees) to the given angle. This shifts the angle halfway around the circle, changing the direction.

Let's take our point, \((-4, \frac{5}{6} \pi)\). Add \(\frac{6}{6} \pi\) to \(\frac{5}{6} \pi\), resulting in \(\frac{11}{6} \pi\). This gives the new coordinates: \((4, \frac{11}{6} \pi)\).

This technique helps in visualizing and plotting points more easily, especially when dealing with negative radii.

Opposite Sign Radius

Finding a coordinate set with an opposite sign radius means changing the radius from negative to positive (or vice versa) while keeping the angle the same.

In our exercise, we have the point \((-4, \frac{5}{6} \pi)\). To find another equivalent set of coordinates with an opposite sign radius, simply change the radius to positive: \((4, \frac{5}{6} \pi)\).

This new set still represents the same point, as the location is mirrored directly across the origin. This method ensures that one can visualize the same point in more than one way, facilitating easier understanding of polar coordinates.