Textbook Question
57–64. Graphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph.
r = sin 3θ
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16. Parametric Equations & Polar Coordinates
Polar Coordinates
2:58 minutesProblem 12.R.29
Textbook Question
27–32. Polar curves Graph the following equations.
r = 3 cos 3θ
Verified step by step guidance1
Understand the given polar equation: \(r = 3 \cos 3\theta\). This represents a polar curve where the radius \(r\) depends on the angle \(\theta\).
Recognize that the equation is of the form \(r = a \cos n\theta\), which typically produces a rose curve with petals. The number of petals depends on \(n\):
- If \(n\) is odd, the rose has \(n\) petals.
- If \(n\) is even, the rose has \$2n$ petals.
Since \(n = 3\) (an odd number), the curve will have 3 petals. The amplitude \$3$ controls the length of each petal.
To graph the curve, create a table of values by choosing several values of \(\theta\) between \$0\( and \(2\pi\), calculate the corresponding \)r$ values using \(r = 3 \cos 3\theta\), and plot the points in polar coordinates.
Connect the plotted points smoothly, noting the symmetry and petal structure, to complete the graph of the rose curve.
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polar Coordinates
Polar coordinates represent points in the plane using a radius and an angle, denoted as (r, θ). Here, r is the distance from the origin, and θ is the angle measured from the positive x-axis. Understanding this system is essential for graphing equations like r = 3 cos 3θ.
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Intro to Polar Coordinates
Graphing Polar Equations
Graphing polar equations involves plotting points by calculating r for various values of θ and then converting these to Cartesian coordinates if needed. Recognizing patterns such as petals or loops helps visualize curves like r = 3 cos 3θ, which produces a rose curve.
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Rose Curves
Rose curves are a family of polar graphs defined by equations of the form r = a cos(nθ) or r = a sin(nθ). The number of petals depends on n: if n is odd, the curve has n petals; if even, it has 2n petals. For r = 3 cos 3θ, the graph has 3 petals.
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