Problem 37 - 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 \(r<0\). $$\left(\frac{3}{4}, \frac{\pi}{6}\right)$$

Step-by-Step Solution

Answer

The two additional pairs of polar coordinates that represent the same point as \((\frac{3}{4}, \frac{\pi}{6})\) are \((\frac{3}{4}, \frac{13\pi}{6})\) and \((- \frac{3}{4}, \frac{7\pi}{6})\).

Step 1: Find Additional Pair with Positive r

Starting with the point \(\left(\frac{3}{4}, \frac{\pi}{6}\right)\), the first task is to find an alternate pair of polar coordinates with positive r, but a different angle. To do this, simply add \(2\pi\) to the given angle as the polar coordinate system is cyclic with period \(2\pi\). It follows that the new pair is \(\left(\frac{3}{4}, \frac{\pi}{6} + 2\pi\right) = \left(\frac{3}{4}, \frac{13\pi}{6}\right)\).

Step 2: Find Additional Pair with Negative r

Next, we derive an alternate pair of polar coordinates with negative r. Achieving this requires switching the direction of the radius by making \(r\) negative and adding \(\pi\) to rotate the angle by a half-turn. Thus, this new pair is \(\left(-\frac{3}{4}, \frac{\pi}{6} + \pi\right) = \left(-\frac{3}{4}, \frac{7\pi}{6}\right)\).

Key Concepts

Angle Conversion

In polar coordinates, the angle, often denoted by \(\theta\), is a crucial component that defines the direction of a point from the origin.
When dealing with polar coordinates, one interesting property is their cyclic nature. They repeat every complete rotation of \(2\pi\) radians or 360 degrees.
This means you can obtain equivalent points by simply adding or subtracting \(2\pi\) to the angle.
When you start with the point \((\frac{3}{4}, \frac{\pi}{6})\), you can find another representation of the same point by adding \(2\pi\).
This results in the angle converting to \(\frac{13\pi}{6}\).

• This understanding aids in generating different angular positions while still referring to the same point in space.
• It also highlights the multiple representations possible for any point in polar coordinates.

Polar coordinates provide flexibility for different geometric interpretations without changing the actual location.

Coordinate Transformation

Transforming between coordinate systems is a key concept in mathematics, particularly when moving from Cartesian to polar coordinates or vice versa.
Polar coordinates \((r, \theta)\) represent points based on a radius \(r\) from the origin and an angle \(\theta\) from a reference direction.
For a given point in polar coordinates, it can be transformed by altering its radius and angle.
For instance, the initial point \((\frac{3}{4}, \frac{\pi}{6})\) can be transformed by changing the angle to \(\frac{13\pi}{6}\), providing a different expression of the same location.

• The transformation offers flexibility in understanding placement from different angular perspectives.
• Understanding this concept is essential for converting between the systems in practical applications, such as physics or engineering.

Remember that even though the representation changes, the spatial location remains the same.

Negative Radius

In polar coordinates, the concept of a negative radius is intriguing.
Normally, a radius \(r\) indicates the distance from the origin to the point, always seen as a positive measure.
However, to explore different positions, we can assign a negative value to \(r\), essentially flipping the direction.
For the point \((\frac{3}{4}, \frac{\pi}{6})\), we attain another equivalent location by setting \(r = -\frac{3}{4}\) and adjusting the angle by \(\pi\).
This transformation gives us the new coordinate \((-\frac{3}{4}, \frac{7\pi}{6})\).

• Essentially, a negative radius makes the point appear on the opposite side of the origin.
• By adding \(\pi\) to the angle, it counterbalances the flipped radius, providing a parallel orientation.

This use of negative radius can simplify certain calculations and offer alternative perspectives in studies involving symmetry.