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Chapter 10: Problem 2

Which polar coordinate pairs label the same point? $$ \begin{array}{lll}{\text { a. }(-2, \pi / 3)} & {\text { b. }(2,-\pi / 3)} & {\text { c. }(r, \theta)} \\ {\text { d. }(r, \theta+\pi)} & {\text { e. }(-r, \theta)} & {\text { f. }(2,-2 \pi / 3)} \\ {\text { g. }(-r, \theta+\pi)} & {\text { h. }(-2,2 \pi / 3)}\end{array} $$

Short Answer

Expert verified
Coordinates a and f represent the same point when converted.

Step by step solution

01

Understand Polar Coordinates

Polar coordinates are given in the form \((r, \theta)\), where \(r\) is the radius (distance from origin) and \(\theta\) is the angle measured from the positive x-axis. A negative radius \(-r\) can be interpreted as moving in the opposite direction at the angle \(\theta + \pi\). Recognizing this helps to understand when two polar coordinates are equivalent.
02

Convert Negative Radius

Coordinate \((-r, \theta)\) is equivalent to \((r, \theta + \pi)\). Therefore, \((-2, \pi/3)\) becomes equivalent to \((2, \pi/3 + \pi)\) which calculates to \((2, 4\pi/3)\).
03

Check Other Equivalences

Checking other patterns, whenever you have \((r, \theta)\), it is equivalent to \((r, \theta + 2n\pi)\) for any integer \(n\). This means you need to add or subtract full circles (i.e., \(2\pi\)) until the angle matches.
04

Check Pair b with Adjustments

For coordinate \((2, -\pi/3)\), add \(2\pi\) to the angle to get it within the standard range: \(-\pi/3 + 2\pi = 5\pi/3\). This converts \((2, -\pi/3)\) to \((2, 5\pi/3)\).
05

Check Pair f with Adjustments

For coordinate \((2, -2\pi/3)\), add \(2\pi\) similarly: \(-2\pi/3 + 2\pi = 4\pi/3\). Thus \((2, -2\pi/3)\) is equivalent to \((2, 4\pi/3)\).
06

Match pairs

Comparing the coordinates, \((-2, \pi/3)\) and \((2, 4\pi/3)\), which came from \((2, -2\pi/3)\), label the same point. Therefore, that what's left as equivalent pairs are a (converted) to f and b (converted) alone matches d.

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Key Concepts

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

Coordinate Equivalence
In the polar coordinate system, points on a plane can be represented in multiple forms based on their coordinates. A coordinate pair \((r, \theta)\) is considered equivalent to another pair if they represent the same point in space. This concept is closely related to how angles and directions work in polar plots.
  • If two polar coordinates have the same magnitude, you can simply change the angle by multiples of full rotations \(2\pi\) to find equivalent pairs, e.g., \((r, \theta) = (r, \theta + 2n\pi)\) where \(n\) is any integer.
  • Converting between equivalent coordinates can also factor in changes to the radius sign, as long as you also appropriately adjust the angle.
Understanding equivalence is crucial when converting coordinates or when you want to find canonical or simplified portrayal of points.
Negative Radius Conversion
Negative radius conversion is an essential trick in working with polar coordinates. A coordinate with a negative radius \((-r, \theta)\) is essentially moving opposite the direction specified by the angle. Thus, it can be rewritten or converted by changing the angle.
  • This happens by adding \(\pi\) radians to the angle because the opposite of an angle \(\theta\) is \(\theta + \pi\).
  • For example, taking a coordinate \((-2, \frac{\pi}{3})\), you can transform it into \((2, \frac{\pi}{3} + \pi)\), eventually simplifying to \((2, \frac{4\pi}{3})\).
This conversion ensures the points mapped by coordinates remain consistent and easily comparable between different forms.
Angle Adjustments
Adjusting angles is pivotal in polar coordinates for helping place angles within a standard range or context. Any given angle \(\theta\) can wrap around the circle, meaning it can be increased or decreased by multiples of \(2\pi\) to result in the same angular placement.
  • This means \((r, \theta)\) is equivalent to \((r, \theta + 2n\pi)\) and helps when you wish to convert negative angles to positive ones or bring large angles into a simpler form.
  • For instance, an angle of \(-\frac{\pi}{3}\) becomes \(\frac{5\pi}{3}\) when \(2\pi\) is added to it, effectively placing it in a more standardized range.
These adjustments assure that angles are uniformly represented and easier to work with computationally and conceptually.
Polar Coordinate System
The polar coordinate system is a method of plotting points in a plane using a radius and an angle. The radius \(r\) indicates the distance from a point to the origin, while the angle \(\theta\) represents how far the point has rotated around the origin from the positive x-axis.
  • Unlike Cartesian coordinates, which use \(x\) and \(y\) values, polar coordinates are inherently circular, offering a distinct perspective useful in fields such as physics or engineering.
  • Understanding the transitions like \((-r, \theta)\) to \((r, \theta + \pi)\) is key in dealing with problems in polar representation, often requiring transformations in the coordinate plane.
Mastering these coordinates aids in efficiently solving problems dealing with rotations, circles, and cycles.

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Most popular questions from this chapter

If lines are drawn parallel to the coordinate axes through a point \(P\) on the parabola \(y^{2}=k x, k>0\) , the parabola partitions the rectangular region bounded by these lines and the coordinate axes into two smaller regions, \(A\) and \(B\) . a. If the two smaller regions are revolved about the \(y\) -axis, show that they generate solids whose volumes have the ratio \(4 : 1 .\) b. What is the ratio of the volumes generated by revolving the regions about the \(x\) -axis? (GRAPH NOT COPY)

Exercises \(29-36\) give the eccentricities of conic sections with one focus at the origin, along with the directrix corresponding to that focus. Find a polar equation for each conic section. $$ e=1 / 5, \quad y=-10 $$

Exercises \(29-36\) give the eccentricities of conic sections with one focus at the origin, along with the directrix corresponding to that focus. Find a polar equation for each conic section. $$ e=1, \quad x=2 $$

Find the areas of the regions Inside the circle \(r=-2 \cos \theta\) and outside the circle \(r=1\)

Find the areas of the regions Inside the lemniscate \(r^{2}=6 \cos 2 \theta\) and outside the circle \(r=\sqrt{3}\)
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