Polar valentine Liz wants to show her love for Jake by passing him a valentine on her graphing calculator. Sketch each of the following curves and determine which one Liz should use to get a heart-shaped curve.
c. r = cos 3θ

Verified step by step guidance

1. Recall that the polar equation \(r = \cos 3\theta\) represents a rose curve, where the number of petals depends on the coefficient of \(\theta\) inside the cosine function.
2. For an equation of the form \(r = \cos n\theta\), if \(n\) is odd, the rose curve has \(n\) petals; if \(n\) is even, it has \$2n\( petals. Here, \)n = 3$, so the curve will have 3 petals.
3. To sketch the curve, consider values of \(\theta\) from \$0\( to \)2\pi\( and calculate \)r\( for key angles such as \)0\(, \)\frac{\pi}{6}\(, \)\frac{\pi}{3}\(, \)\frac{\pi}{2}$, etc., plotting the points in polar coordinates.
4. Plot the points and connect them smoothly to reveal the 3-petal rose shape. Each petal corresponds to the angles where \(\cos 3\theta\) is positive and reaches its maximum.
5. Compare this rose curve to the classic heart-shaped curve (like the cardioid \(r = 1 - \sin \theta\) or \(r = 1 - \cos \theta\)) to determine that \(r = \cos 3\theta\) does not produce a heart shape, so Liz should choose a different polar equation for a heart-shaped valentine.

Key Concepts

Polar Coordinates

Polar coordinates represent points in the plane using a radius and an angle (r, θ) instead of Cartesian coordinates (x, y). The radius r measures the distance from the origin, and θ is the angle from the positive x-axis. Understanding how to plot points using r and θ is essential for sketching polar curves.

Graphing Polar Equations

Graphing polar equations involves calculating r for various values of θ and plotting the corresponding points. Curves like r = cos(nθ) produce rose patterns with petals, where the number of petals depends on n. Recognizing these patterns helps identify shapes such as hearts or flowers.

Rose Curves and Their Properties

Rose curves are polar graphs of the form r = cos(nθ) or r = sin(nθ). If n is odd, the curve has n petals; if even, it has 2n petals. These curves create symmetrical, petal-like shapes, which can resemble hearts or flowers, making them useful for artistic graphing like Liz’s valentine.