Problem 45 - Calculus Early Transcendentals

Question

\(29-48\) Sketch the curve with the given polar equation. $$r=2 \cos (3 \theta / 2)$$

Step-by-Step Solution

Answer

Sketch a polar rose curve with 3 petals.

Step 1: Understanding the Polar Equation

The given polar equation is \( r = 2 \cos \left( \frac{3 \theta}{2} \right) \). Here, \( r \) is the radial distance from the origin and \( \theta \) is the angle in radians. Polar equations often involve trigonometric functions that can create various patterns called 'limaçons', 'rose curves', or others depending upon the function.

Step 2: Identify the Type of Curve

The equation \( r = 2 \cos \left( \frac{3 \theta}{2} \right) \) involves a cosine function and a parameter multiplying \( \theta \). This is related to a rose curve, since we have \( n = \frac{3}{2} \). Rose curves typically have the form \( r = a \cos(n\theta) \) or \( r = a \sin(n\theta) \), where \( n \) determines the number of petals.

Step 3: Determine Number of Petals

For cosine rose curves of the form \( r = a \cos(n\theta) \), if \( n \) is a whole number, \( n \) gives the number of petals if \( n \) is odd, and \( 2n \) if \( n \) is even. Here, \( n = \frac{3}{2} \) is not an integer but acts more like an odd whole number, providing a three-petal rose curve for the completed rotations.

Step 4: Sketching the Curve

To sketch the curve, calculate \( r \) for values of \( \theta \) between \( 0 \) and \( 2\pi \) and plot the points in the polar coordinate plane. For instance, at \( \theta = 0 \), \( r = 2 \cdot 1 = 2 \). At \( \theta = \frac{2\pi}{3} \), the cosine term is zero, giving \( r = 0 \). Continue this process for several values of \( \theta \) to trace the curve.

Key Concepts

Rose Curves

Rose curves are a fascinating type of mathematical curve that can be plotted using polar coordinates. They're named for their petal-like shapes and are usually defined by equations of the form \( r = a \cos(n\theta) \) or \( r = a \sin(n\theta) \). The parameter \( a \) affects the length of the petals, while \( n \) determines the number of petals:

• If \( n \) is odd, the curve will have \( n \) petals.
• If \( n \) is even, the resulting curve will have \( 2n \) petals.

In our focus equation \( r = 2 \cos \left(\frac{3\theta}{2}\right) \), the presence of the fractional \( n = \frac{3}{2} \) creates a visually intriguing pattern with three distinct petal-like loops. The variations in petal numbers spring from the harmonic nature of cosine and sine functions, making these curves not only common in mathematical texts but also visually appealing.

Sketching Polar Curves

Sketching polar curves might initially seem complex, but it becomes much simpler with a structured approach. Polar curves, like rose curves, can be plotted by calculating \( r \), the radial distance, at various angles \( \theta \). Here's how you can get started:

• Select a range of \( \theta \), typically from \( 0 \) to \( 2\pi \).
• Calculate \( r \) for various values of \( \theta \). For instance, begin with known angles like \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \).
• Translate these \( (r, \theta) \) pairs into Cartesian coordinates for easier plotting, using \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \).
• Draw the curve by connecting these points smoothly, appreciating the oscillating nature that defines the petals or loops.

Remember, sketching these curves by hand helps build intuition around their behavior, as you witness how trigonometric functions influence the shape in real-time.

Radial Distance in Polar Coordinates

Understanding radial distance is key to mastering polar coordinates and plotting equations like rose curves. In polar coordinates, the position of a point is determined by two values:

• \( r \): The radial distance from the origin (the center of the coordinate plane).
• \( \theta \): The angle measured from the positive x-axis.

Thus, every point on a polar curve is a combination of these two values, \( (r, \theta) \). The equation you deal with determines \( r \) for various \( \theta \) values. It helps to visualize \( r \) as the length of an imaginary line stretching from the origin to the curve at a specific angle. This dynamic represents why various angles can result in the same \( r \) value, highlighting the symmetrical and often repeating nature of polar curves. By mastering \( r \), you'll better understand how polar equations map out curves in their unique coordinate system.