/*! 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 7 Explain three symmetries in pola... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 7

Explain three symmetries in polar graphs and how they are detected in equations.

Short Answer

Expert verified
Answer: The three symmetries in polar graphs are (a) symmetry about the polar axis, (b) symmetry about the line 胃 = 蟺/2, and (c) symmetry with respect to the origin. They can be detected in equations by the following conditions: 1. Symmetry about the polar axis: If replacing (r, 胃) with (r, -胃) or (-r, 蟺 - 胃) results in the same equation. 2. Symmetry about the line 胃 = 蟺/2: If replacing (r, 胃) with (r, 蟺 - 胃) or (-r, 蟺 + 胃) results in the same equation. 3. Symmetry with respect to the origin: If replacing (r, 胃) with (-r, 胃) or (r, 胃 + 蟺) results in the same equation.

Step by step solution

01

1. Introduction to Polar Graphs and Symmetries

Polar graphs are a way to represent functions and geometric shapes using polar coordinates. In polar coordinates, the location of a point is described by its distance from the origin (r) and its angle from the polar axis (胃). There are three types of symmetries commonly found in polar graphs: (a) symmetry about the polar axis, (b) symmetry about the line 胃 = 蟺/2, and (c) symmetry with respect to the origin.
02

2. Symmetry about the Polar Axis

A polar graph has symmetry about the polar axis if replacing (r, 胃) with (r, -胃) or (-r, 蟺 - 胃) result in the same equation. In other words, if the equation remains unchanged when 胃 is replaced with -胃, or when r and 胃 are both negated, the graph is symmetric about the polar axis.
03

Example 1: Symmetry about the Polar Axis

For example, consider the equation r = 2cos(胃). Replacing 胃 with -胃, we get r = 2cos(-胃), and since cosine is an even function, cos(-胃) = cos(胃), leading to r = 2cos(胃). So, this equation is symmetric about the polar axis. Similarly, replacing r with -r and 胃 with 蟺 - 胃, we get -r = 2cos(蟺 - 胃). Applying the cosine subtraction formula, cos(蟺 - 胃) = -cos(胃), we obtain -r = -2cos(胃), which simplifies back to r = 2cos(胃). This confirms the symmetry about the polar axis.
04

3. Symmetry about the Line 胃 = 蟺/2

A polar graph has symmetry about the line 胃 = 蟺/2 (the vertical axis) if replacing (r, 胃) with (r, 蟺 - 胃) or (-r, 蟺 + 胃) results in the same equation. If the polar equation remains unchanged upon substituting 胃 with 蟺 - 胃 or when r and 胃 are replaced by -r and 蟺 + 胃, respectively, then the graph is symmetric about the line 胃 = 蟺/2.
05

Example 2: Symmetry about the Line 胃 = 蟺/2

For example, consider the equation r = 2sin(胃). Replacing 胃 with 蟺 - 胃, we get r = 2sin(蟺 - 胃). Using the sine subtraction formula, sin(蟺 - 胃) = sin(胃), so r = 2sin(胃). Hence, this equation is symmetric about the line 胃 = 蟺/2. Additionally, replacing r with -r and 胃 with 蟺 + 胃, we get -r = 2sin(蟺 + 胃). The sine function satisfies sin(蟺 + 胃) = -sin(胃), so -r = -2sin(胃). Simplifying, we get r = 2sin(胃), confirming the symmetry about the line 胃 = 蟺/2.
06

4. Symmetry with Respect to the Origin

A polar graph has symmetry with respect to the origin if replacing (r, 胃) with (-r, 胃) or (r, 胃 + 蟺) results in the same equation. If the polar equation remains unchanged when r is replaced with -r or when 胃 is replaced with 胃 + 蟺, then the graph is symmetric with respect to the origin.
07

Example 3: Symmetry with Respect to the Origin

For example, consider the equation r = 胃. Replacing r with -r, we get -r = 胃. This equation is not the same as the original equation, so this graph is not symmetric with respect to the origin by this condition. However, replacing 胃 with 胃 + 蟺, we get r = 胃 + 蟺. By subtracting 蟺 from both sides, we arrive at the equation r - 蟺 = 胃, which describes the same graph as r = 胃. So, this equation is symmetric with respect to the origin. In summary, we have explained three symmetries in polar graphs, symmetry about the polar axis, symmetry about the line 胃 = 蟺/2, and symmetry with respect to the origin, and demonstrated how to detect each of them in polar equations through examples.

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.

Polar Coordinates
Polar coordinates offer a unique system to describe the position of points on a plane. Instead of using the usual x and y Cartesian coordinates, polar coordinates define a point with two values:
  • The distance from the origin, labeled as \( r \).
  • The angle from the polar axis (positive x-axis), denoted by \( \theta \).
This method is especially useful for circular and spiral figures, making complex shapes more intuitive to analyze.
Visualizing concepts like circles with this system can be straightforward. For example, a circle of radius 3 centered at the origin can be represented simply as \( r = 3 \). Understanding polar coordinates serves as a foundation for exploring polar graphs and their properties.
Symmetry
In mathematics, symmetry refers to an attribute where a figure, graph, or equation remains unchanged under certain transformations. Polar graphs, in particular, often exhibit three types of symmetry:
  • Symmetry about the polar axis.
  • Symmetry about the line \( \theta = \pi/2 \).
  • Symmetry with respect to the origin.
Recognizing these symmetries helps in sketching graphs, simplifying equations, and understanding the behavior of polar functions. Once these symmetries are identified, you can predict the shape and position of the graph without plotting every point.
Polar Axis Symmetry
Polar axis symmetry occurs when a polar graph remains unchanged if either:
  • Replace \( (r, \theta) \) with \( (r, -\theta) \).
  • Replace \( (r, \theta) \) with \( (-r, \pi - \theta) \).
For instance, let's consider the equation \( r = 2\cos(\theta) \). By substituting \( \theta \) with \( -\theta \), we see the graph remains the same as cosine is an even function. This confirms the symmetry about the polar axis.
Finding this type of symmetry can simplify graph analysis significantly, particularly for figures aligned with the horizontal axis.
Line 胃 = 蟺/2 Symmetry
Line \( \theta = \pi/2 \) symmetry is present if the graph is unchanged when:
  • Replace \( (r, \theta) \) with \( (r, \pi - \theta) \).
  • Replace \( (r, \theta) \) with \( (-r, \pi + \theta) \).
Take the equation \( r = 2\sin(\theta) \), for example. Substituting \( \theta \) with \( \pi - \theta \) shows the graph is unchanged as sine is odd. This symmetry aligns with the vertical axis and helps in visualizing how the graph appears mirrored across the vertical line.
Origin Symmetry
Origin symmetry indicates the graph looks the same when:
  • Replace \( (r, \theta) \) with \( (-r, \theta) \).
  • Replace \( (r, \theta) \) with \( (r, \theta + \pi) \).
An example can be seen in the equation \( r = \theta \). Checking for origin symmetry, you replace \( \theta \) with \( \theta + \pi \). This leads us back to an equivalent form, suggesting since the graph overlaps onto itself, it has this symmetry.
Understanding this type of symmetry can aid in identifying rotational characteristics of the graph, making complex symmetric figures easier to interpret.

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

A simplified model assumes the orbits of Earth and Mars around the Sun 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 \((3,0) ; \text { both orbit the Sun (at }(0,0))\) 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 Farth is a limacon (Exercise 91 ).

Let \(C\) be the curve \(x=f(t)\), \(y=g(t),\) for \(a \leq t \leq b,\) where \(f^{\prime}\) and \(g^{\prime}\) are continuous on \([a, b]\) and C does not intersect itself, except possibly at its endpoints. If \(g\) is nonnegative on \([a, b],\) then the area of the surface obtained by revolving C about the \(x\)-axis is $$S=\int_{a}^{b} 2 \pi g(t) \sqrt{f^{\prime}(t)^{2}+g^{\prime}(t)^{2}} d t$$. Likewise, if \(f\) is nonnegative on \([a, b],\) then the area of the surface obtained by revolving C about the \(y\)-axis is $$S=\int_{a}^{b} 2 \pi f(t) \sqrt{f^{\prime}(t)^{2}+g^{\prime}(t)^{2}} d t$$ (These results can be derived in a manner similar to the derivations given in Section 6.6 for surfaces of revolution generated by the curve \(y=f(x)\).) Find the area of the surface obtained by revolving the curve \(x=\sqrt{t+1}, y=t,\) for \(1 \leq t \leq 5,\) about the \(y\) -axis.

Equations of the form \(r^{2}=a \sin 2 \theta\) and \(r^{2}=a \cos 2 \theta\) describe lemniscates (see Example 7 ). Graph the following lemniscates. $$r^{2}=4 \sin 2 \theta$$

Consider the following pairs of lines. Determine whether the lines are parallel or intersecting. If the lines intersect, then determine the point of intersection. a. \(x=1+s, y=2 s\) and \(x=1+2 t, y=3 t\) b. \(x=2+5 s, y=1+s\) and \(x=4+10 t, y=3+2 t\) c. \(x=1+3 s, y=4+2 s\) and \(x=4-3 t, y=6+4 t\)

Determine whether the following statements are true and give an explanation or counterexample. a. The area of the region bounded by the polar graph of \(r=f(\theta)\) on the interval \([\alpha, \beta]\) is \(\int_{\alpha}^{\beta} f(\theta) d \theta\). b. The slope of the line tangent to the polar curve \(r=f(\theta)\) at a point \((r, \theta)\) is \(f^{\prime}(\theta)\). c. There may be more than one line that is tangent to a polar curve at some points on the curve.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Pure Maths

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.