27–32. Polar curves Graph the following equations.


r = 3 cos 3θ

Verified step by step guidance

1. Understand the given polar equation: \(r = 3 \cos 3\theta\). This represents a polar curve where the radius \(r\) depends on the angle \(\theta\).
2. 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 \(n\):
3. - If \(n\) is odd, the rose has \(n\) petals.
4. - If \(n\) is even, the rose has \$2n$ petals.
5. Since \(n = 3\) (an odd number), the curve will have 3 petals. The amplitude \$3$ controls the length of each petal.
6. 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.
7. Connect the plotted points smoothly, noting the symmetry and petal structure, to complete the graph of the rose curve.

Key Concepts

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.

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.

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.