27–32. Polar curves Graph the following equations.

r = 3 cos 3θ

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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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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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Introduction to Common Polar Equations

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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