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Chapter 7: Problem 53

Explaining the Concepts Describe the test for symmetry with respect to the polar axis.

Short Answer

Expert verified
A polar equation demonstrates symmetry with the polar axis if substituting 胃 with -胃 in the equation brings you back to the original equation. This concept is used to understand the behavior of graphs and functions in polar coordinates.

Step by step solution

01

Understanding Polar Coordinates

Before diving into the concept of symmetry in polar coordinates, it鈥檚 crucial to understand what polar coordinates are. Polar coordinates system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point (origin, usually denoted as O) and an angle from a reference direction (typically taken as the direction of the positive x-axis). Coordinates of a point P are usually written as (r, 胃), where r is the radius (distance from origin to the point) and 胃 is the angle.
02

Symmetry in Polar Coordinates

Symmetry in polar coordinates can be either symmetric with respect to the polar axis, the pole, or the origin. Symmetry with respect to the polar axis exists if the point P(r, 胃) is on the graph of the equation, then the point P(r, -胃) or P(r, 胃+蟺) is also on the graph.
03

Test for Symmetry in Polar Coordinates

To test a polar equation for symmetry with respect to the polar axis, replace 胃 with -胃 in the equation. If the resulting equation is equivalent to the original equation then the graph is symmetric with respect to the polar axis.

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Key Concepts

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

Symmetry
Symmetry in mathematics describes how one object is mirrored or balanced in relation to another. In polar coordinates, symmetry helps to discover if a shape or graph remains unchanged under certain transformations.
To test for symmetry, particularly with respect to the polar axis, observe these key ideas:
  • Symmetry with the **polar axis** means the shape looks the same on both sides of the axis.
  • A point \(P(r, \theta)\) on the shape should have a mirrored point \(P(r, -\theta)\) also on the shape.
  • Another symmetrical possibility is \(P(r, \theta + \pi)\) which means the shape is reflected across the pole.
Recognizing symmetry helps to simplify analysis and graphing by reducing the variables and potential errors.
Polar Axis
The polar axis is a critical component of the polar coordinate system. Think of it as the equivalent of the x-axis in the Cartesian coordinate system. It's the reference line from which angles are measured.
Key points about the polar axis include:
  • The polar axis usually aligns with the positive x-axis in a rectangular coordinate system.
  • It's the baseline; angles are calculated counterclockwise from this line.
  • When testing for symmetry with respect to the polar axis, you replace \(\theta\) with \(-\theta\) to check if the equation holds true.
This axis provides a foundation for measuring direction and helps determine symmetry in a polar coordinate graph.
Coordinate System
A coordinate system allows us to describe the position of a point on a plane. In polar coordinates, this system uses a combination of distance and angle.
Here's a deeper look into the polar coordinate system compared to the Cartesian system:
  • **Polar coordinates** describe points using \( (r, \theta) \), where \(r\) is the distance from the origin and \(\theta\) is the angle from the polar axis.
  • **Cartesian coordinates** use a grid-based approach with \( (x, y) \) to pinpoint locations.
  • The polar system is simple for circular and radial patterns, while the Cartesian system works well for linear and grid-like structures.
Understanding these systems allows for fluid transition when solving problems that involve polar symmetry, as you can switch between forms depending on what works best for the problem at hand.

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Most popular questions from this chapter

Test for symmetry and then graph each polar equation. $$r=\sin \theta \cos ^{2} \theta$$

Use a graphing utility to graph \(r=\sin n \theta\) for \(n=1,2,3,4,5\) and \(6 .\) Use a separate viewing screen for each of the six graphs. What is the pattern for the number of loops that occur corresponding to each value of \(n ?\) What is happening to the shape of the graphs as \(n\) increases? For each graph, what is the smallest interval for \(\theta\) so that the graph is traced only once?

From a point on level ground 120 feet from the base of a tower, the angle of elevation is \(48.3^{\circ} .\) Approximate the height of the tower to the nearest foot.

You want to fly your small plane due north, but there is a 75-kilometer wind blowing from west to east. a. Find the direction angle for where you should head the plane if your speed relative to the ground is 310 kilometers per hour. b. If you increase your airspeed, should the direction angle in part (a) increase or decrease? Explain your answer.

Use a graphing utility to graph the polar equation. $$r=\cos \frac{3}{2} \theta$$
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