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Chapter 1: Problem 331

For the following exercises, sketch the polar curve and determine what type of symmetry exists, if any. \(r=5 \cos (5 \theta)\)

Short Answer

Expert verified
The curve is a 5-petal rose symmetric about the x-axis.

Step by step solution

01

Understanding the Polar Equation

The given polar equation is \( r = 5 \cos(5\theta) \). This is a rose curve equation, typically of the form \( r = a \cos(n\theta) \) or \( r = a \sin(n\theta) \). In this equation, \( a = 5 \) (the amplitude) and \( n = 5 \) (the number of petals, since \( n \) is odd). This tells us that the curve will have \( n = 5 \) petals.
02

Identifying Symmetry

Since the polar curve is given by \( r = a \cos(n\theta) \), it exhibits symmetry about the polar axis (x-axis). This is because cosine is an even function, and the cosine-based rose curves are generally symmetric about the x-axis.
03

Sketching the Polar Curve

To sketch the rose curve \( r = 5 \cos(5\theta) \), note that the curve completes one cycle when \( \theta \) ranges from \([0, 2\pi]\). The peaks of the petals occur where \( \cos(5\theta) = 1 \), and these values will be equally spaced around the origin. There will be 5 petals since \( n = 5 \). Plot the points for \( \theta = 0, \frac{2\pi}{5}, \frac{4\pi}{5}, \frac{6\pi}{5}, \frac{8\pi}{5} \), and continue to complete the cycle symmetrically due to the periodic nature of the cosine function.

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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 a fascinating concept as it reveals inherent patterns within polar curves. By understanding symmetry, sketching and analyzing these curves becomes much simpler. There are three main types of symmetry to consider:
  • Polar Axis Symmetry (x-axis): If a polar equation remains unchanged when you replace \(\theta\) with \(-\theta\), then the curve is symmetric about the polar axis, also known as the x-axis.
  • Line \(\theta = \frac{\pi}{2}\) Symmetry (y-axis): For symmetry about this line, replacing \(r\) with \(-r\) in the equation results in the same graph.
  • Origin Symmetry: This occurs when replacing both \(r\) with \(-r\) and \(\theta\) with \(-\theta\) results in the same graph.
For the rose curve given by \(r = 5 \cos(5\theta)\), the symmetry about the polar axis is evident because cosine is an even function.
The curve looks identical if you flip it around the x-axis, making the sketch predictable and visually even.
Rose Curves
A rose curve is a beautiful and intriguing type of polar graph. These curves resemble the petals of a flower, and their form is dictated by the equation \(r = a \cos(n\theta)\) or \(r = a \sin(n\theta)\).
  • Number of Petals: In the case of \(r = a \cos(n\theta)\), the value of \(n\) determines the number of petals. When \(n\) is odd, the curve has \(n\) petals. For even \(n\), the curve displays \(2n\) petals.
  • Amplitude: The parameter \(a\) sets the length of each petal from the origin to the furthest point, showcasing the amplitude.
  • Patterns and Symmetry: Rose curves often exhibit polar axis symmetry, especially when cosine is involved, leading to symmetrical and aesthetically pleasing graphs.
In the example \(r = 5 \cos(5\theta)\), there are 5 petals due to the odd \(n\), each extending out to a length of 5, unerringly arranged around the center, reflecting perfect symmetry.
Sketching Polar Graphs
Sketching polar graphs can be tackled systematically with a blend of understanding the equations and visual symmetries. Here's a step-by-step guide:
  • Identify the Type of Curve: Determine whether the equation represents a circle, rose curve, or another polar form.
  • Calculate Key Points: For rose curves like \(r = a \cos(n\theta)\), determine where max and min values occur by examining when \(\cos(n\theta) = 1\) or \(-1\).
  • Determine Symmetry: Use known symmetries to sketch half or a portion of the graph, then reflect or extend as needed.
  • Plot Cycles: Recognize complete cycles and range necessary to cover \(\theta\), like \([0, 2\pi]\) for a full image of rose curves.
When sketching \(r = 5 \cos(5\theta)\), consider the symmetry about the x-axis. Start by locating the angles where the petals peak and space them evenly across the given range of \(\theta\).
This method yields a complete, accurate, and neatly presented polar graph.

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