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Chapter 10: Problem 71

Sketch the following sets of points \((r, \theta)\). \(|\theta| \leq \frac{\pi}{3}\)

Short Answer

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Question: Sketch the region in polar coordinates with the constraint \(|\theta| \leq \frac{\pi}{3}\). Answer: To create a sketch of the polar coordinates with the given constraint, follow these steps: 1. List the constraint: \(|\theta| \leq \frac{\pi}{3}\) or \(-\frac{\pi}{3} \leq \theta \leq \frac{\pi}{3}\). 2. Sketch a horizontal line to represent \(\theta=0\). 3. Fill the region between \(\theta = -\frac{\pi}{3}\) and \(\theta = \frac{\pi}{3}\) by covering all possible positive values of \(r\). 4. Label the sketch to clearly indicate the constraint, \(|\theta| \leq \frac{\pi}{3}\).

Step by step solution

01

List the constraint

We are given the set of points \((r, \theta)\) such that \(|\theta| \leq \frac{\pi}{3}\). This constraint means that the angle \(\theta\) will be in the range \(-\frac{\pi}{3} \leq \theta \leq \frac{\pi}{3}\).
02

Sketch a horizontal line

The set of points with angle \(\theta\) in the given range will lie between the lines \(\theta = -\frac{\pi}{3}\) and \(\theta = \frac{\pi}{3}\). Therefore, we begin by sketching the lines that represent these angles. Since there is no constraint on the radial coordinate \(r\), choose some arbitrary positive values for \(r\), such as \(r=1\), \(r=2\), and \(r=3\).
03

Sketch the angle ranges

With the horizontal line drawn, represent the angular range by filling the region between the angle \(\theta = -\frac{\pi}{3}\) and \(\theta = \frac{\pi}{3}\). Choose points with angle \(\theta\) within this range, and make sure to cover the entire range of possible \(r\) values.
04

Label the sketch

Finally, label the sketch of the region in polar coordinates with the given constraint \(|\theta| \leq \frac{\pi}{3}\). By following these steps, you should have a diagram displaying the set of points \((r, \theta)\) with the constraint \(|\theta| \leq \frac{\pi}{3}\), covering the whole range of \(\theta\) values and all possible positive \(r\) values.

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Key Concepts

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

Understanding Angle Constraints in Polar Coordinates
When working with polar coordinates, an angle constraint defines the range of angles for which we can consider a certain set of points. In the provided exercise, the constraint is given as \(|\theta| \leq \frac{\pi}{3}\). This can be interpreted as the angular interval from \(-\frac{\pi}{3}\) to \(\frac{\pi}{3}\). To visualize this, imagine slicing a pie. The sections between \(-\frac{\pi}{3}\) and \(\frac{\pi}{3}\) represent the allowable angles.

These constraints help us focus only on the specific wedges of the full circle that fit the conditions. We use radians for angle measurement, where \(\pi\) is equivalent to 180 degrees. Thus, angles ranging from \(-60\degree\) to \(60\degree\) are acceptable.
- **Imagining these slices**: Think of the constraint as slices in a circle between \(-60\degree\) and \(60\degree\). Both positive and negative angles are inclusive.- **Usage in problems**: This helps in narrowing down the points to be considered when looking at certain segments around the origin.
Understanding Radial Coordinate
The radial coordinate, represented as \(r\), specifies the distance from the origin to a point in polar coordinates. In the given exercise, the radial coordinate doesn鈥檛 have specific constraints, so \(r\) can be any positive number.

The radial coordinate reminds us how we view the world from a center outward. In contexts like these:
  • **Origin reference**: Think of the origin as your point of view. The farther from the center, the bigger \(r\) gets.
  • **No boundary restriction**: Unlike the angle constraints, \(r\) is only dependent on your needs, varying from zero up to infinity.
To better understand:- Consider \(r = 1, 2, 3,...\) as steps getting visually farther from a fixed point. Thus, no constraint broadens the scope, making your field of possible locations grow exponentially.
Visualization in Polar Coordinates
Visualization in polar coordinates offers an intuitive understanding of how points are plotted and regions defined.In this exercise:

We start by plotting lines for \(\theta = -\frac{\pi}{3}\) and \(\theta = \frac{\pi}{3}\). These lines act like the boundary markers.
  • **Sketching**: Draw these as straight lines emanating from the origin at specified angles.
  • **Filling in points**: The area between these lines is the zone where all allowable points will appear.
This visualization grants insights into possible distances (\(r\)) and angular positions.Deep understanding through visualization helps grasp:- How varying \(r\) and \(\theta\) build the image of an object or set up regions.- Empowers you to use geometry to capture all possible points in the given angle.

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Most popular questions from this chapter

A simplified model assumes that the orbits of Earth and Mars are circular with radii of 2 and \(3,\) respectively, and that Earth completes one orbit in one year while Mars takes two years. When \(t=0,\) Earth is at (2,0) and Mars is at (3,0) both orbit the Sun (at (0,0) ) in the counterclockwise direction. The position of Mars relative to Earth is given by the parametric equations \(x=(3-4 \cos \pi t) \cos \pi t+2, \quad y=(3-4 \cos \pi t) \sin \pi t\) a. Graph the parametric equations, for \(0 \leq t \leq 2\) b. Letting \(r=(3-4 \cos \pi t),\) explain why the path of Mars relative to Earth is a lima莽on (Exercise 89).

Water flows in a shallow semicircular channel with inner and outer radii of \(1 \mathrm{m}\) and \(2 \mathrm{m}\) (see figure). At a point \(P(r, \theta)\) in the channel, the flow is in the tangential direction (counterclockwise along circles), and it depends only on \(r,\) the distance from the center of the semicircles. a. Express the region formed by the channel as a set in polar coordinates. b. Express the inflow and outflow regions of the channel as sets in polar coordinates. c. Suppose the tangential velocity of the water in \(\mathrm{m} / \mathrm{s}\) is given by \(v(r)=10 r,\) for \(1 \leq r \leq 2 .\) Is the velocity greater at \(\left(1.5, \frac{\pi}{4}\right)\) or \(\left(1.2, \frac{3 \pi}{4}\right) ?\) Explain. d. Suppose the tangential velocity of the water is given by \(v(r)=\frac{20}{r},\) for \(1 \leq r \leq 2 .\) Is the velocity greater at \(\left(1.8, \frac{\pi}{6}\right)\) or \(\left(1.3, \frac{2 \pi}{3}\right) ?\) Explain. e. The total amount of water that flows through the channel (across a cross section of the channel \(\theta=\theta_{0}\) ) is proportional to \(\int_{1}^{2} v(r) d r .\) Is the total flow through the channel greater for the flow in part (c) or (d)?

Consider the curve \(r=f(\theta)=\cos a^{\theta}-1.5\) where \(a=(1+12 \pi)^{1 /(2 \pi)} \approx 1.78933\) (see figure). a. Show that \(f(0)=f(2 \pi)\) and find the point on the curve that corresponds to \(\theta=0\) and \(\theta=2 \pi\) b. Is the same curve produced over the intervals \([-\pi, \pi]\) and \([0,2 \pi] ?\) c. Let \(f(\theta)=\cos a^{\theta}-b,\) where \(a=(1+2 k \pi)^{1 /(2 \pi)}, k\) is an integer, and \(b\) is a real number. Show that \(f(0)=f(2 \pi)\) and that the curve closes on itself. d. Plot the curve with various values of \(k\). How many fingers can you produce?

Graph the following spirals. Indicate the direction in which the spiral is generated as \(\theta\) increases, where \(\theta>0 .\) Let \(a=1\) and \(a=-1\). Hyperbolic spiral: \(r=a / \theta\)

A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties. The lines tangent to the endpoints of any focal chord of a parabola \(y^{2}=4 p x\) intersect on the directrix and are perpendicular.
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