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Chapter 6: Problem 33

Test for symmetry and then graph each polar equation. $$r \cos \theta=-3$$

Short Answer

Expert verified
The equation \(r \cos \theta=-3\) is symmetric about the x-axis. The graph is a vertical line at \(x = -3\).

Step by step solution

01

Test for symmetry about the x-axis

Replace \(\theta\) in the equation with \(-\theta\). If the equation remains unchanged, then it's symmetric about the x-axis. So we replace \(\theta\) with \(-\theta\) in the given equation \(r \cos \theta=-3\), which results: \(r \cos(-\theta) =-3\). Since \(\cos(-\theta) = \cos \theta\), the equation \(r \cos \theta=-3\) remains unchanged. Therefore, the equation does show symmetry about the x-axis.
02

Test for symmetry about the y-axis

Replace \(r\) and \(\theta\) in the equation with \(-r\) and \(\pi-\theta\). If the equation remains unchanged, then it's symmetric about the y-axis. So we replace \(r\) and \(\theta\) with \(-r\) and \(\pi-\theta\) in the original equation. The equation becomes \(-r \cos (\pi-\theta) =-3\). Using the property that \(\cos(\pi - \theta) = - \cos\theta\), our equation becomes \(r \cos \theta = 3\), which is not the same as the original equation. Therefore, the equation does not show symmetry about the y-axis.
03

Test for symmetry about the origin

Replace \(r\) and \(\theta\) in the equation with \(-r\) and \(-\theta\). If the equation remains unchanged, then it's symmetric about the origin. So we replace \(r\) and \(\theta\) with \(-r\) and \(-\theta\) in the original equation. The equation becomes \(-r \cos(-\theta) =-3\). Using the property that \(\cos(-\theta) = \cos\theta\), the equation becomes \(-r \cos \theta = - 3\), which is not the same as the original equation. Therefore, the equation does not show symmetry about the origin.
04

Draw the Graph

As \(r \cos \theta=-3\) is a form of \(x = -3\) (since \(r \cos \theta = x\) in polar coordinates), we have a vertical line through \(x = -3\), which is symmetric about the x-axis.

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Key Concepts

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

Symmetry
Symmetry in polar coordinates refers to different ways a graph can mirror itself or appear unchanged upon various transformations. Knowing these symmetries helps in understanding and simplifying the graphing process of polar equations.
  • X-axis symmetry: Symmetry with respect to the x-axis occurs when the graph looks the same above and below the x-axis. To test this, replace \(\theta\) with \(-\theta\) in the equation. If the equation holds true, it confirms x-axis symmetry.
  • Y-axis symmetry: For y-axis symmetry, the graph mirrors itself across the y-axis. Replace \(r\) with \(-r\) and \(\theta\) with \(\pi - \theta\). The equation should remain unchanged if y-axis symmetry exists.
  • Origin symmetry: This occurs when the graph remains the same if it is rotated 180 degrees around the origin. We check this by replacing \(r\) with \(-r\) and \(\theta\) with \(-\theta\).
These symmetry tests help to determine the nature of the graph, simplifying plotting by indicating repeated patterns in specific sections of the polar plane.
Graphing Polar Equations
Graphing polar equations involves plotting points in a coordinate system defined by a radius and angle, rather than the typical x and y coordinates. This unique system provides different insights and curves compared to Cartesian graphing.
  • Understanding Polar Coordinates: In polar coordinates, each point on a graph is defined by a pair \((r, \theta)\), where \(r\) denotes the distance from the origin, and \(\theta\) represents the angle from the positive x-axis. This often results in circular or spiral forms unlike those seen in traditional graphs.
  • Plotting Points: Start by calculating and plotting points for various angles. Use known symmetries to reduce the workload.
  • Connecting the Dots: Use the plotted points to sketch out the curve. The understanding of symmetries can help decide if certain sections should mirror others.
By mastering graphing in this system, one gains the ability to visualize complex equations in simpler, more intuitive forms. Polar graphs offer insights into aspects like periodicity and rotational behavior that may be less evident in Cartesian graphs.
X-axis Symmetry
Understanding x-axis symmetry in polar graphs plays a crucial role in simplifying the graphing process. When a polar graph exhibits x-axis symmetry, it remains unchanged or mirrors itself along the x-axis when the angle \(\theta\) is negated.
  • Testing for X-axis Symmetry: Replace \(\theta\) with \(-\theta\) in your polar equation. For example, in the equation \(r \cos \theta = -3\), substituting \(-\theta\) yields \(r \cos(-\theta) = -3\). Since \(\cos(-\theta) = \cos \theta\), the equation is identical, thus confirming x-axis symmetry.
  • Graphical Implications: With confirmed x-axis symmetry, one needs to plot points only for one half of the graph. These points are mirrored across the x-axis, simplifying plotting. This is a significant time-saver.
X-axis symmetry ensures the graph is balanced across the x-axis, providing a chance to identify patterns and simplify graphic representation. Understanding this concept aids in reducing complexity and enhancing accuracy in plotting polar equations.

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Most popular questions from this chapter

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. $$(8.3,4.6)$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When I convert an equation from polar form to rectangular form, the rectangular equation might not define \(y\) as a function of \(x .\)

Use the vectors \(\mathbf{u}=a_{1} \mathbf{i}+b_{1} \mathbf{j}, \quad \mathbf{v}=a_{2} \mathbf{i}+b_{2} \mathbf{j}, \quad\) and \(\quad \mathbf{w}=a_{3} \mathbf{i}+b_{3} \mathbf{j}\) to prove the given property. $$(c \mathbf{u}) \cdot \mathbf{v}=c(\mathbf{u} \cdot \mathbf{v})$$

a. Does \((4,-1)\) satisfy \(x+2 y=2 ?\) b. Does \((4,-1)\) satisfy \(x-2 y=6 ?\)
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