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Chapter 8: Problem 78

Which equation has a graph that is a four-leaved rose? a. \(r=3 \cos 4 \theta\) b. \(r=5 \sin 2 \theta\) c. \(r=2+2 \cos \theta\) d. \(r=3+5 \sin \theta\)

Short Answer

Expert verified
The correct equation is \(r=5 \sin 2 \theta\).

Step by step solution

01

Identify the Polar Equation

Rose curves are patterns or graphs of certain polar equations given by either the form \(r = a \cos(n\theta)\) or \(r = a \sin(n\theta)\). These graphs have petal-like patterns.
02

Recognize the Symmetry

Rose curves described by \(r = a \cos(n\theta)\) or \(r = a \sin(n\theta)\) have different symmetries. If \(n\) is even, the rose has \(2n\) petals; if odd, exactly \(n\) petals.
03

Analyze Option a

The equation \(r=3 \cos 4 \theta\) fits the form for rose curves. Here, \(n=4\), and thus it will have \(2 \times 4 = 8\) petals.
04

Analyze Option b

The equation \(r=5 \sin 2 \theta\) also fits the rose curve form, where \(n=2\), leading to \(2 \times 2 = 4\) petals, matching the description of a four-leaved rose.
05

Analyze Option c

The equation \(r=2+2 \cos \theta\) describes a lima莽on with an inner loop, not a rose curve.
06

Analyze Option d

The equation \(r=3+5 \sin \theta\) also forms a lima莽on, not a rose curve.
07

Conclusion

The correct polar equation that renders a four-leaved rose is \(r=5 \sin 2 \theta\), which is the option b.

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Key Concepts

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

Rose Curves
Rose curves are fascinating graphs in polar coordinates, resembling petal-like formations. These arise from polar equations of the form \(r = a \cos(n\theta)\) or \(r = a \sin(n\theta)\). These equations inherently determine the number of petals based on the integer \(n\):
  • When \(n\) is even, the rose curve will display \(2n\) petals.
  • When \(n\) is odd, it will show exactly \(n\) petals.
By carefully analyzing these patterns, one can decipher the elegant structures and predict the number of petals exhibited by the curve. Matching these properties with the given options, option b, \(r=5 \sin 2 \theta\), describes a four-petal rose curve, as \(2 \times 2 = 4\). This characteristic feature is key to recognizing rose curves in polar equations.
Symmetry in Polar Graphs
Symmetry plays a significant role in simplifying the graphing of polar equations. In rose curves such as \(r = a \cos(n\theta)\) and \(r = a \sin(n\theta)\), the symmetry hints how the curves will manifest themselves around the polar center:
  • Equations of the form \(r = a \cos(n\theta)\) are symmetric about the polar axis (x-axis in the polar graph).
  • Equations of the form \(r = a \sin(n\theta)\) exhibit symmetry about the line \(\theta = \frac{\pi}{2}\) (y-axis in the polar graph).
Understanding these symmetrical properties can assist in both sketching the curves and identifying their orientation quickly. Symmetry reduces the complexity, helping to predict parts of the curve without detailed calculations for each point.
Lima莽on Curves
Lima莽on curves are another intriguing type of curve in polar coordinates and come from the equations \(r = a + b \cos(\theta)\) or \(r = a + b \sin(\theta)\). Depending on the values of \(a\) and \(b\), lima莽on curves can display different forms:
  • If \(b > a\), there is an inner loop within the curve.
  • If \(b = a\), the curve will form a cardioid, which is heart-shaped.
  • If \(b < a\), the lima莽on will exhibit a dimpled form.
In the context of the exercise, both options c and d, \(r=2+2 \cos \theta\) and \(r=3+5 \sin \theta\), respectively, result in lima莽on curves rather than rose curves. Recognizing this distinction is crucial for correctly identifying types of polar graphs and understanding their structure.

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