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Chapter 6: Problem 35

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

Short Answer

Expert verified
The polar equation \(r = \cos(\theta/2)\) is symmetric with respect to the polar axis but not with respect to the pole or the line \(\theta = \pi / 2\). The graph is a cardioid shape.

Step by step solution

01

Testing for Symmetry

To find if the function is symmetrical with respect to the origin or the polar axis, the following rules can be applied: \n1. Replace \(\theta\) with \(-\theta\) - if the equation remains the same, it is symmetrical about the polar axis.\n2. Replace \(\theta\) with \(\pi - \theta\) - if there is no change, then symmetry with respect to the pole exists.\n3. Replace r and \(\theta\) with -r and \(\pi + \theta\) - if the result remains the same, the graph shows symmetry with respect to the line \(\theta = \pi / 2\). \nNow perform these operations on our equation \(r = \cos(\theta/2)\).
02

Analyzing Test Results for Symmetry

1. Replace \(\theta\) with \(-\theta\): \(r = \cos(-\theta/2)\), which is equal to \(r = \cos(\theta/2)\). So, it is symmetrical with respect to the polar axis.\n2. Replace \(\theta\) with \(\pi - \theta\): \(r = \cos((\pi - \theta)/2)\), which is not equal to the original equation, hence no symmetry exists with respect to the pole.\n3. Replace r and \(\theta\) with -r and \(\pi + \theta\): \(r = \cos((\pi + \theta)/2)\), and this is also not equal to the original equation, hence no symmetry with respect to the line \(\theta = \pi / 2\).
03

Graphing the Equation

To draw the graph for this equation you need to create a table with values of \(\theta\) ranging from \(0\) to \(2\pi\). Calculate the corresponding r-values using the equation \(r = \cos(\theta/2)\). Plot these points (\(r, \theta\)) on the polar coordinate plane. Connect these points to form a curve. A polar graph paper can be used for better accuracy.

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Key Concepts

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

Symmetry in Polar Coordinates
Symmetry in polar coordinates is crucial for understanding the shape and behavior of polar graphs. It helps in simplifying the graphing process by predicting unseen parts of the graph based on its symmetrical properties.

There are three tests to determine symmetry:
  • **Polar Axis Symmetry:** Replace \(\theta\) with \(-\theta\). If the equation is unaffected, it is symmetric about the polar axis.
  • **Pole Symmetry:** Substitute \(\theta\) with \(\pi - \theta\). An unchanged equation indicates symmetry with respect to the pole.
  • **Line \(\theta = \pi / 2\) Symmetry:** Replace \(r, \theta\) with \(-r, \pi + \theta\). If unchanged, the graph is symmetric about the line \(\theta = \pi / 2\).
Understanding these symmetries aids in anticipating how the polar graph will be structured, which can simplify the graphing process.
Polar Coordinates
Polar coordinates provide a different way of describing points in a plane compared to the familiar Cartesian coordinates. In the polar coordinate system, each point is determined by a distance from a reference point and an angle from a reference direction.

The reference point is known as the pole (similar to the origin in Cartesian coordinates), and the reference direction is the polar axis (akin to the positive x-axis). A polar coordinate is written as \ \ \(r, \theta\)\, where:
  • **\(r\)** is the radial coordinate: the distance from the pole.
  • **\(\theta\)** is the angular coordinate: the angle from the polar axis.
This coordinate system is exceptionally useful in situations involving circular or angular paths, such as waves or planetary motion.
Graphing Polar Equations
Graphing polar equations involves plotting points on a polar grid based on their \(r\) and \(\theta\) values and connecting these points to reveal the shape of the graph.

Here are steps to make the process easier:
  • Select a range for \(\theta\), often from \(0\) to \(2\pi\), although sometimes it might differ based on the equation's nature.
  • Calculate corresponding \(r\) values for several angles within your selected range using your polar equation.
  • Plot the points \((r, \theta)\) on polar graph paper for precision and clarity.
  • Connect the points carefully, observing any genuine curves or patterns.
Using polar graph paper provides pre-drawn concentric circles and lines radiating from the center, making it easier to plot the polar coordinates accurately.

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Most popular questions from this chapter

Polar coordinates of a point are given. Use a graphing utility to find the rectangular coordinates of each point to three decimal places. $$(-4,1.088)$$

Verify the identity: $$\sin ^{2} x \tan ^{2} x+\cos ^{2} x \tan ^{2} x=\sec ^{2} x-1$$

Convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. $$r=-4 \sin \theta$$

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$\left(-6, \frac{3 \pi}{2}\right)$$

Convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. $$r=8 \cos \theta+2 \sin \theta$$
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