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Chapter 8: Problem 238

Graph the polar inequality \(0 \leq \theta \leq \frac{\pi}{3}, r<2\)

Short Answer

Expert verified
The graph is a sector of a circle with radius less than 2, between angles 0 and \(\frac{\pi}{3}\).

Step by step solution

01

Understanding the Polar Coordinates

In polar coordinates, every point on the plane is determined by a radius \(r\) and an angle \(\theta\). Here, the inequality \(0 \leq \theta \leq \frac{\pi}{3}\) describes a sector of the circle. The angle starts at \(0\) (positive x-axis) and ends at \(\frac{\pi}{3}\) (60 degrees from the positive x-axis).
02

Identifying the Circle

The condition \(r < 2\) specifies the region inside a circle of radius 2. This means all points in this sector of the circle should have a radius less than 2.
03

Combining Conditions

The solution to the inequality is the set of points that meet both conditions. This means the region inside the circle with radius 2, but only for angles between \(0\) and \(\frac{\pi}{3}\). It's a pie-shaped region extending from the origin to the circle's edge, but cutting off at a radius of 2.

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Key Concepts

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

Polar Inequality
A polar inequality is a restriction on the range of possible values in polar coordinates. It involves conditions on either the angle \( \theta \), the radius \( r \), or both, which define specific regions on the polar plane.
In the context of the exercise, we have the polar inequality \(0 \leq \theta \leq \frac{\pi}{3}, r<2\). This inequality specifies two separate conditions that govern the positions of points.
  • The range for \( \theta \), from \(0\) to \(\frac{\pi}{3}\), indicates the sector of the polar plane where the points can reside.
  • The condition \( r < 2 \) restricts the distance from the origin, ensuring all points are within a circular boundary with radius 2.
By satisfying both, we locate the specific region meant by this polar inequality; it's a sector-like shape cut from a circle.
Circle Sector
A circle sector, commonly referred to as a 'pie slice', is part of a circle defined by a central angle. In polar coordinates, a sector is distinguished by a range of \( \theta \) values.
In our exercise example, the sector is determined by \(0 \leq \theta \leq \frac{\pi}{3}\). This means:
  • Start from the positive x-axis, which is \(\theta = 0\).
  • Move in a counter-clockwise direction up to an angle of \(\frac{\pi}{3}\), equivalent to 60 degrees.
This forms a wedge emanating from the origin, shaping a sector of a circle.
Such sectors are significant in problems where you're interested in subsets of a circle defined by specific angle intervals.
Note, however, that the full sector here would be an infinite wedge without any restriction on \( r \), spanning outward indefinitely.
Radius and Angle
In polar coordinates, each point on the plane is best described by its radius \( r \) and angle \( \theta \). These form the basis for understanding positions and shapes on the polar plane.
The radius \( r \):
  • Denotes the distance from the origin to a point.
  • Is always a non-negative value.
  • In this exercise with \( r < 2 \), the points cannot reach beyond a circle of radius 2.
The angle \( \theta \):
  • Indicates the direction from the positive x-axis.
  • Measured in radians or degrees, with \( \pi \) radians equating to 180 degrees.
By employing both radius and angle, you can navigate the entire polar plane. It's essential to remember that while the radius determines proximity to the origin, the angle defines direction. This pair works harmoniously to describe and restrict regions as specified by conditions or inequalities such as those seen in this activity.

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Most popular questions from this chapter

For the following exercises, find a unit vector in the same direction as the given vector. $$\boldsymbol{b}=-3 \boldsymbol{i}-\boldsymbol{j}$$

For the following exercises, find the powers of each complex number in polar form. Find \(z^{2}\) when \(z=5 \operatorname{cis}\left(\frac{3 \pi}{4}\right)\)

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For the following exercises, look at the graphs of each of the four parametric equations. Although they look unusual and beautiful, they are so common that they have names, as indicated in each exercise. Use a graphing utility to graph each on the indicated domain. A hypocycloid: \(\left\\{\begin{array}{l}{x(t)=6 \sin t+2 \sin (6 t)} \\\ {y(t)=6 \cos t-2 \cos (6 t)}\end{array} \text { on the domain }[0,2 \pi]\right.\)

Graph the set of parametric equations and find the Cartesian equation: \(\left\\{\begin{array}{l}{x(t)=-2 \sin t} \\ {y(t)=5 \cos t}.\end{array}\right.\)
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