/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 58 Sketch the curve with the polar ... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 58

Sketch the curve with the polar equation. \(r=1-2 \cos \theta \quad\) (lima莽on)

Short Answer

Expert verified
To sketch the curve with the polar equation \(r = 1 - 2\cos\theta\) (lima莽on), we first find the pole by setting \(r = 0\) and solving for \(\theta\). This gives us \(\theta = \pm\frac{\pi}{3}\) and \(\theta = \pm\frac{5\pi}{3}\), indicating an inner loop. We then plot points for various values of \(\theta\) and connect them smoothly to sketch the lima莽on, which consists of an outer loop and an inner loop based on the pole locations.

Step by step solution

01

Identify Key Features of the Graph

Begin by finding the pole, which is the point where the lima莽on touches the origin (i.e., when \(r = 0\)). To find the pole, set \(r = 0\) and solve for \(\theta\): \(0 = 1 - 2\cos\theta\) \(2\cos\theta = 1\) \(\cos\theta =\frac{1}{2}\) \(\theta = \pm\frac{\pi}{3}\) and \(\theta = \pm\frac{5\pi}{3}\) Since there are two values for \(\theta\) when \(r = 0\), the lima莽on will have an inner loop.
02

Plot the Points

Now, plot points for various values of \(\theta\) on the polar coordinate system to find the shape of the graph: \(\theta = 0\) gives \(r = 1-2\cos(0)=1-2(1)=-1\) \(\theta = \frac{\pi}{6}\) gives \(r = 1-2\cos\left(\frac{\pi}{6}\right)=1-2\left(\frac{\sqrt{3}}{2}\right)=-\sqrt{3}+1\) \(\theta = \frac{\pi}{3}\) gives \(r = 1-2\cos\left(\frac{\pi}{3}\right)=1-2\left(\frac{1}{2}\right)=0\) (pole) \(\theta = \frac{\pi}{2}\) gives \(r = 1-2\cos\left(\frac{\pi}{2}\right)=1-2(0)=1\) \(\theta = \frac{2\pi}{3}\) gives \(r = 1-2\cos\left(\frac{2\pi}{3}\right)=1-2\left(-\frac{1}{2}\right)=2\) (inner loop) \(\theta = \frac{5\pi}{6}\) gives \(r = 1-2\cos\left(\frac{5\pi}{6}\right)=1-2\left(-\frac{\sqrt{3}}{2}\right)=\sqrt{3}+1\) \(\theta = \pi\) gives \(r = 1-2\cos(\pi)=1-2(-1)=3\) \(\theta = \frac{7\pi}{6}\) gives \(r = 1-2\cos\left(\frac{7\pi}{6}\right)=1-2\left(-\frac{\sqrt{3}}{2}\right)=\sqrt{3}+1\) \(\theta = \frac{5\pi}{3}\) gives \(r = 1-2\cos\left(\frac{5\pi}{3}\right)=1-2\left(\frac{1}{2}\right)=0\) (pole) \(\theta = \frac{11\pi}{6}\) gives \(r = 1-2\cos\left(\frac{11\pi}{6}\right)=1-2\left(\frac{\sqrt{3}}{2}\right)=-\sqrt{3}+1\) \(\theta = 2\pi\) gives \(r = 1-2\cos(2\pi)=1-2(1) = -1\)
03

Sketch the Graph

Now, plot these points on a polar coordinate system and connect them smoothly to sketch the lima莽on. The graph will have an outer loop and an inner loop based on the pole locations.

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Key Concepts

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

尝颈尘补莽辞苍
The lima莽on is a fascinating type of polar graph that results from equations of the form \( r = a \, + \, b \cos \theta \) or \( r = a \, + \, b \sin \theta \). It is named after the French word for "snail" due to its characteristic shape, which often resembles a coil or loop.

There are various types of lima莽ons, depending on the values of \( a \) and \( b \):
  • If \( b < a \), the curve is a dimpled lima莽on.
  • If \( b = a \), the curve is a cardioid, which looks like a heart.
  • If \( b > a \), the curve has an inner loop.
In the exercise, with the equation \( r = 1 - 2 \cos \theta \), the curve has an inner loop due to the magnitude relationship \( b > a \), creating a unique shape with two distinct loops.
Graphing Polar Coordinates
Graphing polar coordinates involves plotting points on a circular grid, distinct from the Cartesian coordinate system. Points in polar coordinates are determined by a radius \( r \) and an angle \( \theta \).

To plot a polar point:
  • Start from the polar axis (analogous to the positive x-axis in Cartesian coordinates).
  • Rotate counter-clockwise by an angle \( \theta \).
  • Measure the radius or distance \( r \) from the pole, the origin of the polar coordinate system.
For positive \( r \) values, move outward from the pole, whereas negative \( r \) values mean you plot the point in the opposite direction from \( \theta \). This method showcases how polar coordinates effectively represent circular and rotational graphs, such as the lima莽on.
Polar Coordinate System
The polar coordinate system is a two-dimensional plane where each point is defined by a distance \( r \) and an angle \( \theta \). It is especially useful for graphs with circular symmetry and when dealing with periodic functions.

Key features include:
  • The pole, the origin of the system with coordinates \( (0, 0) \).
  • The polar axis, equivalent to the positive x-axis in Cartesian coordinates.
  • Angles measured in radians, looping back to zero after \( 2\pi \) radians.
This system is preferred for functions involving rotations and angles, as it simplifies the graphing process compared to Cartesian coordinates, making it easier to visualize complex curves like lima莽ons.
Cosine Function
The cosine function is pivotal in defining polar equations, particularly those of the form \( r = a \, + \, b \cos \theta \). In our equation \( r = 1 - 2 \cos \theta \), the cosine function is responsible for the periodic changes in \( r \) as \( \theta \) varies, creating the loops characteristic of the lima莽on.

Cosine measures the x-coordinate of a point on the unit circle correlating with the angle \( \theta \).

Key properties include:
  • Periodicity with a cycle of \( 2\pi \), meaning it repeats every \( 2\pi \) radians.
  • Values ranging from -1 to 1, affecting the radius in polar equations.
Understanding the cosine function's behavior helps anticipate the peaks and troughs of the lima莽on graph, effectively guiding its accurate plotting.

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Most popular questions from this chapter

Find the area of the region described. The inner loop of the lima莽on \(r=1+2 \cos \theta\)

Find the area of the region that lies outside the first curve and inside the second curve. $$ r=1+\cos \theta, \quad r=3 \cos \theta $$

a. Let \(f\) be a function with a continuous derivative in an interval \([\alpha, \beta]\). If the graph \(C\) of \(r=f(\theta)\) is traced exactly once as \(\theta\) increases from \(\alpha\) to \(\beta\), show that the rectangular coordinates of the centroid of \(C\) are $$ \bar{x}=\frac{\int_{\alpha}^{\beta} r \cos \theta \sqrt{\left(r^{\prime}\right)^{2}+r^{2}} d \theta}{\int_{\alpha}^{\beta} \sqrt{\left(r^{\prime}\right)^{2}+r^{2}} d \theta} $$ and $$ \bar{y}=\frac{\int_{\alpha}^{\beta} r \sin \theta \sqrt{\left(r^{\prime}\right)^{2}+r^{2}} d \theta}{\int_{\alpha}^{\beta} \sqrt{\left(r^{\prime}\right)^{2}+r^{2}} d \theta} $$ Hint: See the directions for Exercises 47 and 48 in Section \(5.7\). b. Use the result of part (a) to find the centroid of the upper semicircle \(r=a\), where \(a>0\) and \(0 \leq \theta \leq \pi\).

Sketch the curve, and find the area of the region enclosed by it. $$ r=2(1-\cos \theta) $$

Find the length of the given curve. $$ r=\sec \theta ; \quad 0 \leq \theta \leq \frac{\pi}{3} $$
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