Problem 27 - The Calculus with Analytic Geometry

Question

Draw a sketch of the graph of the given equation.\(r=4-3 \cos \theta\) (limaçon)

Step-by-Step Solution

Answer

Plot points for specific \(\theta\) values and draw a smooth curve. Example points: (1, 0), (4, \(\frac{\pi}{2}\)), (7, \(\pi\)).

Step 1: Understand the Given Equation

The given equation is in polar coordinates: \(r=4-3 \cos \theta\). This is an equation of a limaçon. Limaçons are a type of polar curve which can have different shapes, depending on the parameter values.

Step 2: Identify Key Points

Calculate the values of \(r\) for key angles \(\theta\). For example: \(\theta = 0\): \(r = 4 - 3 \cos(0) = 4 - 3 = 1\) \(\theta = \frac{\pi}{2}\): \(r = 4 - 3 \cos(\frac{\pi}{2}) = 4 - 3 \cdot 0 = 4\) \(\theta = \pi\): \(r = 4 - 3 \cos(\pi) = 4 - 3 \cdot (-1) = 4 + 3 = 7\)Repeat for other important angles if needed.

Step 3: Plot Key Points

Plot the calculated points on polar coordinate paper or use a Cartesian coordinate system in a polar context. For example, plot (1, 0), (4, \(\frac{\pi}{2}\)), and (7, \(\pi\)) on the polar graph.

Step 4: Draw the Limaçon Curve

Connect the points smoothly, considering the symmetry and shape of a limaçon. It usually has an inner loop if the magnitude of the coefficients results in one. The form 4 - 3 cos(θ) typically will not have a loop since 4 > 3.

Key Concepts

Limaçon

A limaçon is a type of polar graph defined by equations of the form \( r = a + b \, \cos(\theta) \) or \( r = a - b \, \cos(\theta) \). Depending on the values of \( a \) and \( b \), the shape of the limaçon can vary significantly. Some common variations include:

• When \( a = b \), the curve features an inner loop.
• When \( a > b \), the curve resembles a dimpled shape without an inner loop.
• If \( b > a \), the limaçon will have an outer loop.

The given equation, \( r = 4 - 3 \, \cos(\theta) \), falls under the category where \( a > b \), indicating that the limaçon will not have an inner loop but will instead have a dimpled heart-like shape. Observing how varying the values of \( a \) and \( b \) affects the curve's structure helps understand different forms of limaçons better.

Polar Graph

A polar graph represents data points in terms of the polar coordinate system, where each point is determined by a radius \( r \) and an angle \( \theta \). Unlike Cartesian coordinates that use \( (x, y) \) pairs to locate points, polar coordinates use \( (r, \theta) \) pairs.

Here's how to understand the polar graph:

• \( r \): The distance from the origin to the plotted point.
• \( \theta \): The angle measured in radians or degrees from the positive x-axis (counter-clockwise).

For our limaçon equation, we calculated the radius \( r \) for various angles \( \theta \). These angles form key points that help draw the overall shape of the curve. Key points like \( (1, 0) \), \( (4, \frac{\pi}{2}) \), and \( (7, \pi) \) provide essential anchors to sketch the polar graph accurately.

Key Angles

Key angles play a crucial role in mapping the graph of a polar equation. For complex curves like the limaçon, computing the radius \( r \) at specific angles helps in plotting critical points accurately. These angles are generally:

• \( \theta = 0 \)
• \( \theta = \frac{\pi}{2} \)
• \( \theta = \pi \)
• \( \theta = \frac{3\pi}{2} \)
• Any additional angles that define significant points on the curve

For the equation \( r = 4 - 3 \, \cos(\theta) \), the computed radii for these angles helped in locating the key points (1, 0), (4, \frac{\pi}{2}), and (7, \pi). Once these points were plotted, connecting them smoothly resulted in the dimpled heart-shaped limaçon, accurately reflecting the polar equation. Measuring radius at key angles ensures that the graph is well-defined and anchors important features of the curve.