Problem 12
Question
Identify the symmetries of the curves in Exercises \(1-12 .\) Then sketch the curves. $$ r^{2}=-\cos \theta $$
Step-by-Step Solution
Verified Answer
The curve shows symmetry about the polar axis and the pole, and it forms a double spiral.
1Step 1: Understand the Problem Statement
We are given the polar equation \(r^2 = -\cos \theta\). Our task is to identify its symmetries and sketch the curve based on its properties.
2Step 2: Test for Symmetry about the Polar Axis
To test for symmetry about the polar axis, replace \(\theta\) with \(-\theta\). If the resulting equation remains unchanged, it has polar axis symmetry.Start with the original equation: \[ r^2 = -\cos \theta \]Replace \(\theta\) with \(-\theta\): \[ r^2 = -\cos(-\theta) = -\cos \theta \].The equation is unchanged, so the curve is symmetric about the polar axis.
3Step 3: Test for Symmetry about the Line \(\theta = \frac{\pi}{2}\)
To check symmetry about the line \(\theta = \frac{\pi}{2}\), replace \(\theta\) with \(\pi - \theta\). If the equation remains the same, it is symmetric about this line.Replace \(\theta\) in the original equation:\[ r^2 = -\cos(\pi - \theta) = \cos \theta \].This results in \(r^2 = \cos \theta\), which is not equivalent to the original equation, so there is no symmetry about the line \(\theta = \frac{\pi}{2}\).
4Step 4: Test for Symmetry about the Pole
For symmetry about the pole (origin), replace \(r\) by \(-r\). If the equation remains unchanged, then symmetry exists.Original equation: \[ r^2 = -\cos \theta \].Replacing \(r\) with \(-r\) gives: \[ (-r)^2 = r^2 = -\cos \theta \].The equation remains the same, indicating symmetry about the pole.
5Step 5: Sketch the Curve
The equation \(r^2 = -\cos \theta\) implies we are plotting points where \(r^2\) can be negative, which typically suggest imaginary values—impossible for real planes. Therefore, this curve exists if and only if \(\cos \theta\) is positive. The curve traces values against real negative \(\theta\) from \([-\frac{\pi}{2}, \frac{\pi}{2}]\), looking like a "double spiral" about the origin.
Key Concepts
SymmetryCurve SketchingPolar Equation Analyzing
Symmetry
Symmetry is an integral part of understanding polar curves. Symmetry can simplify graphing by reducing the amount of work needed. It provides insights into the behavior of a curve. This exercise involves looking at different types of symmetry: symmetry about the polar axis, symmetry about the line \(\theta = \frac{\pi}{2}\), and symmetry about the pole.
When testing for symmetry about the polar axis, replace \(\theta\) with \(-\theta\). If the equation remains unchanged, the curve is symmetric about this axis. In our equation \(r^2 = -\cos \theta\), substituting \(\theta\) with \(-\theta\) gives the same equation, indicating symmetry about the polar axis.
To assess symmetry about the line \(\theta = \frac{\pi}{2}\), replace \(\theta\) with \(\pi - \theta\). The transformation results in \(r^2 = \cos \theta\), which differs from the original equation, indicating no symmetry here.
For pole symmetry, replace \(r\) with \(-r\). The square operation means \((-r)^2 = r^2\), confirming pole symmetry. Recognizing these symmetries helps predict the shape and orientation of the curve.
When testing for symmetry about the polar axis, replace \(\theta\) with \(-\theta\). If the equation remains unchanged, the curve is symmetric about this axis. In our equation \(r^2 = -\cos \theta\), substituting \(\theta\) with \(-\theta\) gives the same equation, indicating symmetry about the polar axis.
To assess symmetry about the line \(\theta = \frac{\pi}{2}\), replace \(\theta\) with \(\pi - \theta\). The transformation results in \(r^2 = \cos \theta\), which differs from the original equation, indicating no symmetry here.
For pole symmetry, replace \(r\) with \(-r\). The square operation means \((-r)^2 = r^2\), confirming pole symmetry. Recognizing these symmetries helps predict the shape and orientation of the curve.
Curve Sketching
Sketching curves in polar coordinates can be challenging without practice. Unlike Cartesian equations, polar equations depend on varying angles and radii. To sketch the curve of \(r^2 = -\cos \theta\), consider the symmetry and possible values of \(r\).
First, consider the implications of \(r^2\) being negative since square values are typically non-negative. Thus, this polar equation has solutions only when \(-\cos \theta\) is non-negative—or \(\cos \theta\) is negative.
This restricts us to angles where \(\theta\) yields negative \(\cos \theta\), specifically from angles \(\theta = \frac{\pi}{2}\) to \(\theta = \frac{3\pi}{2}\) in the conventional coordinate system.
The curve possesses polar axis symmetry and pole symmetry, resulting in a symmetric figure often visible as a double spiral. To accurately sketch it, plot points within the mentioned range and observe how they mirror due to detected symmetries.
First, consider the implications of \(r^2\) being negative since square values are typically non-negative. Thus, this polar equation has solutions only when \(-\cos \theta\) is non-negative—or \(\cos \theta\) is negative.
This restricts us to angles where \(\theta\) yields negative \(\cos \theta\), specifically from angles \(\theta = \frac{\pi}{2}\) to \(\theta = \frac{3\pi}{2}\) in the conventional coordinate system.
The curve possesses polar axis symmetry and pole symmetry, resulting in a symmetric figure often visible as a double spiral. To accurately sketch it, plot points within the mentioned range and observe how they mirror due to detected symmetries.
Polar Equation Analyzing
Analyzing polar equations involves more than just manipulating symbols—it requires understanding how equations translate to geometric shapes. The equation \(r^2 = -\cos \theta\) presents an opportunity to engage deeply with such analysis.
To begin with, notice the use of \(r^2\). This suggests a focus on the distance from the origin rather than directly the direction, which implicates only positive distances when solving polar equations.
Given the nature of \(-\cos \theta\), finding values that make this condition true primes us. Here, \(\cos \theta\) must be negative, implying that \(\theta\) is an angle where the cosine graph dips below the x-axis. This often occurs from \(\theta = \frac{\pi}{2}\) to \(\theta = \frac{3\pi}{2}\).
This curve is special in polar coordinates due to its resulting shape—a double spiral, leading into itself as it complies with the constraints given. Understanding such intricacies enhances the ability to work with more complex polar equations.
To begin with, notice the use of \(r^2\). This suggests a focus on the distance from the origin rather than directly the direction, which implicates only positive distances when solving polar equations.
Given the nature of \(-\cos \theta\), finding values that make this condition true primes us. Here, \(\cos \theta\) must be negative, implying that \(\theta\) is an angle where the cosine graph dips below the x-axis. This often occurs from \(\theta = \frac{\pi}{2}\) to \(\theta = \frac{3\pi}{2}\).
This curve is special in polar coordinates due to its resulting shape—a double spiral, leading into itself as it complies with the constraints given. Understanding such intricacies enhances the ability to work with more complex polar equations.
Other exercises in this chapter
Problem 12
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