In Exercises 55-58, use a graphing utility to find the...

In Exercises $55-58,$ use a graphing utility to find the smallest value of $\theta$ max, with $\theta \min =0,$ such that the entire curve is graphed exactly once without retracing.
$$r=3 \cos (2 \theta)$$

Key Concepts

Curve Tracing and Retracing

When graphing polar curves, it is important to choose a range for the angle ? that traces the entire curve exactly once. If the chosen interval is too wide, parts of the curve may be traced more than once. Determining the minimal interval needed to trace the entire curve without retracing involves understanding the symmetry and periodicity inherent in the polar equation.

Periodicity in Polar Equations

Many polar equations, especially those defining rose curves, are periodic. The periodic nature of these curves means that after a certain interval in ?, the curve starts to repeat. Understanding the period of a polar equation is crucial as it allows one to determine the smallest interval over which the entire curve is traced without retracing any part of it.

Rose Curves

Rose curves are a family of curves defined by equations of the form r = a cos(k?) or r = a sin(k?). They are characterized by their petal-like shapes, and the number of petals depends on the parameter k. When k is even, the curve exhibits 2k petals; when k is odd, it has k petals.

Polar Coordinate System

The polar coordinate system uses a radius and an angle to specify the position of points in the plane. This framework is fundamental for representing curves where the relationship between the radius r and the angle ? is given, and it often simplifies the representation of curves that have rotational symmetry or periodic properties.

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