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Chapter 6: Problem 12

Plot the point given in polar coordinates and find two additional polar representations of the point, using \(-2 \pi<\theta<2 \pi\). \(\left(-3, \frac{11 \pi}{6}\right)\)

Short Answer

Expert verified
The equivalent polar representations of \(-3, \frac{11 \pi}{6}\) with \(-2 \pi<\theta<2 \pi\) are \(3, -\frac{5\pi}{6}\) and \(3, \frac{7 \pi}{6}\).

Step by step solution

01

Draw the Given Point

Draw the point with polar coordinates \(-3, \frac{11 \pi}{6}\). Notice that it has a negative radius, which means we should visualize the radius extending into the opposite quadrant. The angle is \(\frac{11 \pi}{6}\), which is just short of \(2 \pi\) and it's slightly more than one and a half a full rotation from the origin.
02

Convert to Equivalent Representation

We first convert the original representation to a more conventional one (i.e., with a positive radius). We know that a point can be represented as \(r, \theta+2 \pi k\), where \(r\) is the radial coordination, \(\theta\) is the polar angle, and \(k\) is an integer. Therefore, an equivalent representation with a positive radius would be \(3, \theta + \pi\) because we kept the angle the same but extended the radius in the opposite direction. In this case, \(\theta + \pi = \frac{11 \pi}{6}+ \pi = \frac{17 \pi}{6}\). But we need to restrict the range of the angle to \(-2\pi<\theta<2\pi\), or else improve it to this range, so the first equivalent representation would be \(3, -\frac{5\pi}{6}\).
03

Find Another Representation

Since a point can be represented as \(r, \theta+2 \pi k\), we can find another representation by adding \(2 \pi\) to the angle. This will keep the point plotted in the same position. So if we add \(2 \pi\) to -\(\frac{5 \pi}{6}\), we get \(-\frac{5 \pi}{6} + 2 \pi = \frac{7 \pi}{6}\). Therefore, another equivalent representation of the point is \(3, \frac{7 \pi}{6}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Negative Radius in Polar Coordinates
In polar coordinates, the radius is a crucial component and is usually denoted by \( r \). It represents the distance from the origin (the center of the polar grid) to the point. Unlike Cartesian coordinates, in polar coordinates, the radius can also be negative. This means that instead of marking the distance directly in the direction of the polar angle, you mark it in the opposite direction.
  • For example, a negative radius like \(-3\) in polar coordinates would mean you measure 3 units away from the pole but in the opposite direction of the angle given.
  • This essentially places the point in the diametrically opposite quadrant in the polar plane.
Understanding that the radius can be negative allows for a better appreciation of the flexibility and uniqueness of polar coordinates.
Decoding the Polar Angle
Polar angle, represented by \( \theta \), is the angle between the positive x-axis and the line segment from the origin to the point. This angle is measured in radians and can be in the range from \(-\pi\) to \(\pi\) (for one loop around the circle), or, as in our exercise, extended to \(-2\pi < \theta < 2\pi\) (for slightly more or less than one loop around).
  • For instance, the angle \( \frac{11 \pi}{6} \) falls in the fourth quadrant because it is less than \( 2 \pi \) but greater than \( \pi \).
  • Angles in polar coordinates are periodic, meaning adding or subtracting \(2 \pi\) (a full circle) results in an equivalent angle visually on the plane.
This cyclic nature of angles is significant when finding equivalent representations of points because a shift like \( \theta + 2\pi \) repositions the angle back to the same location on the circle.
Equivalent Representations in Polar Coordinates
Finding equivalent representations in polar coordinates relies on the cyclical nature of angles and the flexibility in radius direction. A single point in polar coordinates can have many valid representations due to these characteristics.
  • If you flip the sign of the radius from negative to positive, you must adjust the angle by \( \pi \) to compensate for the direction change. For example, if the original point is \((-3, \frac{11\pi}{6})\), its equivalent representation with a positive radius is \( (3, -\frac{5\pi}{6}) \).
  • Additionally, adding or subtracting \(2\pi\) from the angle yields another valid representation, as the angle essentially rotates back to its original direction. Thus, \( (3, \frac{7\pi}{6}) \) is another form.
These properties of polar coordinates mean that each point has an infinite number of equivalent expressions, providing flexibility and multiple ways to express location on a polar plot.

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