/*! 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 9 Graph the following equations. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 9

Graph the following equations. $$ r=\frac{2}{1-\cos (\theta)} $$

Short Answer

Expert verified
The graph is a lima莽on with an inner loop, symmetric about the polar axis.

Step by step solution

01

Identify the Polar Equation

The equation given is a polar equation expressed as \( r = \frac{2}{1 - \cos(\theta)} \). This can be recognized as the equation of a conic section in polar form, specifically a type of lima莽on.
02

Determine the Form of the Lima莽on

The equation \( r = \frac{2}{1 - \cos(\theta)} \) can be compared with the general form for lima莽ons: \( r = \frac{ed}{1 - e\cos(\theta)} \) where \( e = 1 \) when \( d = 2 \). This indicates a lima莽on with an inner loop.
03

Identify Key Angles and Points

Since the lima莽on has an inner loop, it鈥檚 important to identify key angles: \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi \). For example, at \( \theta = \pi \), \( r = \frac{2}{1 - (-1)} = 1 \).
04

Sketch the Complete Graph

Using the key angles, plot points for \( r \) and sketch the lima莽on. The graph will have a loop extending inward towards the pole \((r = 0)\) for certain angles, and an outer loop reaching \( r \rightarrow \infty \) as \( \theta \rightarrow 0 \).
05

Check for Symmetry and Completion

Since the equation involves \( \cos(\theta) \), it is symmetric about the horizontal axis (polar axis). Ensure the graph displays this symmetry before finalizing.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Understanding Lima莽ons
Lima莽ons are fascinating curves in the study of polar coordinates. Their distinctive shape varies based on certain parameters in their equations. A lima莽on can appear as a simple loop or have an inner loop, like the equation given in the original exercise. The standard form of a lima莽on is:
  • \[ r = rac{ed}{1 - e\cos(\theta)} \]
Here, the parameter \( e \) is known as the eccentricity, and it determines the shape of the lima莽on. For example:
  • If \( e = 1 \), it forms a lima莽on with an inner loop as seen in our exercise.
  • If \( e < 1 \), it produces a dimpled lima莽on.
  • When \( e > 1 \), the curve becomes a lima莽on without any loops.
Knowing these variations helps in predicting the shape and understanding the appearance of the graph.
Exploring Conic Sections
Conic sections include a variety of shapes such as circles, ellipses, parabolas, and hyperbolas. They are generated by slicing through a cone at different angles and positions. In polar coordinates, conic sections can be represented in a variety of forms. The standard polar equation for conic sections is:
  • \[ r = \frac{ed}{1 - e\cos(\theta)} \]
This equation not only describes lima莽ons but also other conic sections:
  • When \( e = 0 \), the graph is a circle.
  • When \( 0 < e < 1 \), it's an ellipse.
  • If \( e = 1 \), it鈥檚 a parabola.
  • For \( e > 1 \), it forms a hyperbola.
Recognizing these patterns is beneficial when analyzing graphs in polar coordinates.
Symmetry in Polar Graphs
Symmetry is a key element when graphing polar equations. It simplifies the drawing by reducing the number of points we need to plot. Different types of symmetry include:
  • Symmetry about the x-axis (polar axis): Indicates that the graph looks the same above and below this axis. This type was observed in the original exercise.
  • Symmetry about the y-axis: The graph is mirrored at \( \theta = \frac{\pi}{2} \).
  • Symmetry about the origin: The plot exhibits rotational symmetry.鈥�
To determine symmetry, substituting variables can often provide confirmation. Checking whether the given equation satisfies any of these conditions can help simplify plotting.
Graphing Polar Equations
When graphing polar equations, it helps to follow a systematic approach. For equations like lima莽ons, these steps can guide the process:
  • Identify the type of conic section, like a lima莽on.
  • Determine the characteristics of the curve by inspecting parameters of the equation.
  • Identify key angles such as \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi \). Calculating corresponding \( r \) values at these angles provides crucial points.
  • Check for symmetry to reduce plotting work.
  • Finally, plot key points and sketch the curve, ensuring the graph respects identified symmetries.
Using these steps, drawing even complex graphs like lima莽ons becomes manageable.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In Exercises \(1-20\), plot the set of parametric equations by hand. Be sure to indicate the orientation imparted on the curve by the parametrization. $$ \left\\{\begin{array}{l} x=2 t \\ y=t^{2} \end{array} \text { for }-1 \leq t \leq 2\right. $$

For the given vector \(\vec{v}\), find the magnitude \(\|\vec{v}\|\) and an angle \(\theta\) with \(0 \leq \theta<360^{\circ}\) so that \(\vec{v}=\|\vec{v}\|\langle\cos (\theta), \sin (\theta)\rangle\) (See Definition 11.8.) Round approximations to two decimal places. $$ \vec{v}=\langle-114.1,42.3\rangle $$

In Exercises \(1-20\), plot the set of parametric equations by hand. Be sure to indicate the orientation imparted on the curve by the parametrization. $$ \left\\{\begin{array}{l} x=3 \cos (t) \\ y=3 \sin (t) \end{array} \text { for } 0 \leq t \leq \pi\right. $$

We know that \(|x+y| \leq|x|+|y|\) for all real numbers \(x\) and \(y\) by the Triangle Inequality established in Exercise 36 in Section 2.2. We can now establish a Triangle Inequality for vectors. In this exercise, we prove that \(\|\vec{u}+\vec{v}\| \leq\|\vec{u}\|+\|\vec{v}\|\) for all pairs of vectors \(\vec{u}\) and \(\vec{v}\). (a) (Step 1) Show that \(\|\vec{u}+\vec{v}\|^{2}=\|\vec{u}\|^{2}+2 \vec{u} \cdot \vec{v}+\|\vec{v}\|^{2}\). 6 (b) (Step 2) Show that \(|\vec{u} \cdot \vec{v}| \leq\|\vec{u}\|\|\vec{v}\| .\) This is the celebrated Cauchy-Schwarz Inequality. (Hint: To show this inequality, start with the fact that \(|\vec{u} \cdot \vec{v}|=|\|\vec{u}\|\|\vec{v}\| \cos (\theta)|\) and use the fact that \(|\cos (\theta)| \leq 1\) for all \(\theta\).) (c) (Step 3) Show that \(\|\vec{u}+\vec{v}\|^{2}=\|\vec{u}\|^{2}+2 \vec{u} \cdot \vec{v}+\|\vec{v}\|^{2} \leq\|\vec{u}\|^{2}+2|\vec{u} \cdot \vec{v}|+\|\vec{v}\|^{2} \leq\|\vec{u}\|^{2}+\) \(2\|\vec{u}\|\|\vec{v}\|+\|\vec{v}\|^{2}=(\|\vec{u}\|+\|\vec{v}\|)^{2}\) (d) (Step 4) Use Step 3 to show that \(\|\vec{u}+\vec{v}\| \leq\|\vec{u}\|+\|\vec{v}\|\) for all pairs of vectors \(\vec{u}\) and \(\vec{v}\). (e) As an added bonus, we can now show that the Triangle Inequality \(|z+w| \leq|z|+|w|\) holds for all complex numbers \(z\) and \(w\) as well. Identify the complex number \(z=a+b i\) with the vector \(u=\langle a, b\rangle\) and identify the complex number \(w=c+d i\) with the vector \(v=\langle c, d\rangle\) and just follow your nose!

The HMS Sasquatch leaves port with bearing \(\mathrm{S} 20^{\circ} \mathrm{E}\) maintaining a speed of 42 miles per hour (that is, with respect to the water). If the ocean current is 5 miles per hour with a bearing of \(\mathrm{N} 60^{\circ} \mathrm{E}\), find the HMS Sasquatch's true speed and bearing. Round the speed to the nearest mile per hour and express the heading as a bearing, rounded to the nearest tenth of a degree.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.