Problem 11 - Precalculus with Limits

Question

In Exercises 5-18, plot the point given in polar coordinate sand find two additional polar representations of the point, using \(-2\pi<\theta<2\pi\). \(\left(-2, \dfrac{2\pi}{3}\right)\)

Step-by-Step Solution

Answer

The two additional polar representations of \(\left(-2, \dfrac{2\pi}{3}\right)\) are \(\left(2, \dfrac{5\pi}{3}\right)\) and \(\left(-2, \dfrac{8\pi}{3}\right)\).

Step 1: Plotting the Polar Coordinate

The given polar coordinates \(\left(-2, \dfrac{2\pi}{3}\right)\) will be plotted on a polar coordinate graph. Since the radius is negative, the plotted point counts for the angle \(\dfrac{2\pi}{3}\) will be in the opposite direction i.e. it will add/subtract \(\pi\) from the given angle.

Step 2: Finding Additional Representations

The conversion rule among polar coordinates is \( (r, θ) = (−r, θ ± π) = (r, θ ± 2π)\). To find the first additional representation, add \(\pi\) to the \(θ\), so it becomes \(\left(2, \dfrac{2\pi}{3} + \pi\right) = \left(2, \dfrac{5\pi}{3}\right)\), where radius is taken as positive. To find the second additional representation, add \(2\pi\) to the \(θ\) on the originally given polar coordinate. This second additional representation becomes \(\left(-2, \dfrac{2\pi}{3} + 2\pi\right) = \left(-2, \dfrac{8\pi}{3}\right)\).

Step 3: Verifying the Results

Lastly, plot these points on the polar coordinate graph and verify if they match with initial polar coordinate point i.e., \(\left(-2, \dfrac{2\pi}{3}\right)\), to ensure correctness.

Key Concepts

Angle Conversion

Understanding angle conversion is crucial when dealing with polar coordinates. In polar coordinates, a point is represented by a radius and an angle \((r, \theta)\). The angle \(\theta\) is often expressed in radians. To convert an angle, you may add or subtract multiples of \(2\pi\).

This is because \(2\pi\) radians are equivalent to a full circle or 360 degrees. By adjusting the angle by \(2\pi\), the radial direction remains unchanged, but it provides a different representation of the same point:

• Addition of \(\pi\): Involves moving to the opposite direction on the polar graph; it works when the radius is negative.
• Addition or subtraction of \(2\pi\): Keeps the point's location constant, just provides another numerical representation.

Using this will aid in finding additional polar coordinates when the angle is constrained, like in the exercise \((-2 < \theta < 2\pi)\). Practicing various conversions will enhance your fluency in navigating polar graphs.

Negative Radius

A negative radius is a unique feature in polar coordinates. It's different from Cartesian coordinates that we are more familiar with. A negative radius can be imagined as moving in the opposite direction of the given angle \(\theta\).

Using polar coordinates, the point \((-2, \frac{2\pi}{3})\) implies moving an 'inward' distance of 2 along the line at an angle of \(\frac{2\pi}{3}\). When the radius \(r\) is negative, the direction is reversed. Therefore, adding \(\pi\) to the angle can give the equivalent point with a positive radius.

This is because:

• Negative radius acts as a signal to turn around the angle.
• The angle \(\theta + \pi\) repositions the point to the opposite side of the origin.

This flip can be helpful to translate into a positive radius format. Grasping this concept will simplify working with complex polar transformations.

Polar Representation

Polar representation of a point is expressed as \((r, \theta)\) in the polar coordinate system. Unlike Cartesian coordinates, which use \(x\) and \(y\), polar coordinates use a distance \(r\) and an angle \(\theta\) from a reference direction.

It's possible to express a single point in polar coordinates in multiple ways. This is achieved through:

• Altering the radius from negative to positive by modifying the angle\(\theta + \pi\).
• By adding or subtracting \(2\pi\) to the angle \(\theta\) without altering the radius.

The exercise showcases these methods by converting \((-2, \frac{2\pi}{3})\) into alternative forms like \((2, \frac{5\pi}{3})\) and \((-2, \frac{8\pi}{3})\).
This shows that, through mathematical transformations, the same point can appear different numerically. Mastery of polar representation involves understanding this variability, and assessing the geometrical implications of these number changes.