Problem 60 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 45-78: Sketch the graph of the polar equation. $$ r=2 \sqrt{3}-4 \cos \theta $$

Step-by-Step Solution

Answer

It's a limaçon with an inner loop, symmetric about the polar axis.

Step 1: Understand Polar Equations

Polar equations are expressed in the form \( r = f(\theta) \), where \( r \) is the radius or distance from the origin and \( \theta \) is the angle. In this case, the equation is \( r = 2 \sqrt{3} - 4 \cos \theta \). The aim is to understand how \( r \) changes with \( \theta \) and sketch the graph based on this behavior.

Step 2: Identify the Type of Curve

The given equation is of the form \( r = a + b \cos \theta \). This is a polar form of a circle known as a limaçon. Depending on the relationship between \( a \) and \( b \), it might have an inner loop, be a cardioid, be a dimpled limaçon, or be a convex limaçon. Here, \( a = 2 \sqrt{3} \) and \( b = 4 \). Evaluate \( a, b \) relationship: \( 2\sqrt{3} < 4 \), suggesting it is a limaçon with an inner loop.

Step 3: Determine Key Points and Symmetry

Determine the symmetry and key points by evaluating \( r \) at \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \) and \( 2\pi \) to understand the curve's orientation. \( r(0) = 2 \sqrt{3} - 4\), \( r\left(\frac{\pi}{2}\right) = 2 \sqrt{3} \), \( r(\pi) = 2 \sqrt{3} + 4 \), \( r\left(\frac{3\pi}{2}\right) = 2 \sqrt{3} \). These calculations show that the figure is symmetric with respect to the polar axis.

Step 4: Check Loop Existence

The procedure includes solving \( r = 0 \) to understand \( \theta \) where the inner loop begins or ends. Set \( 2 \sqrt{3} = 4 \cos \theta \), \( \cos \theta = \frac{\sqrt{3}}{2} \), implying \( \theta = \frac{\pi}{6} \) and \( \theta = \frac{11\pi}{6} \). Between these angles, \( r \) will be negative, indicating an inner loop where the radius temporarily reads negative.

Step 5: Sketch the Graph

Using the analyzed data, sketch a limaçon with an inner loop. Begin tracing at \( \theta = 0 \) progressing in a counter-clockwise direction. The curve moves inward creating a loop between \( \theta = \frac{\pi}{6} \) and \( \theta = \frac{11\pi}{6} \). At \( \theta = 0 \) and \( \pi \), the curve reaches significant radii, indicating points of larger circles.

Key Concepts

Polar Coordinates

Polar coordinates provide an alternative method to represent points in a plane by using a radius and angle as opposed to traditional Cartesian coordinates. Instead of a point being determined by x and y values, polar coordinates use the following:

• \( r \): The radius, which is the distance from the origin (the pole).
• \( \theta \): The angle, expressed in radians, measuring from the positive x-axis.

This system is particularly useful for graphing curves that naturally incorporate angles and distances, such as circles and spirals.
To convert between polar and Cartesian coordinates:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

This allows for flexibility in graphing and is particularly efficient when dealing with phenomena that are best described in terms of rotation and radial distance from a fixed point.

Limaçon Curve

A limaçon, pronounced 'lee-mah-sohn', is a type of polar curve whose general form is expressed as \( r = a + b \cos \theta \) or \( r = a + b \sin \theta \). These curves can manifest in several forms:

• **Inner loop**: Occurs when \( |b| > |a| \).
• **Cardioid**: Happens when \( |b| = |a| \).
• **Dimpled**: Forms when \( |a| > |b| > 0 \).
• **Convex**: Appears when \( |a| \geq |b| \).

In this exercise, the equation \( r = 2 \sqrt{3} - 4 \cos \theta \) is identified as a limaçon curve with an inner loop due to the relationship \( 2\sqrt{3} < 4 \). This means that there are values of \( \theta \) for which the radius \( r \) becomes negative, creating an inner loop in the graph.

Graphing Techniques

When graphing polar equations such as limaçons, several techniques aid in understanding and sketching the curve:

• **Symmetry identification**: This assists in minimizing computational work by showing mirrored behaviors around axes or the pole.
• **Key points evaluation**: Calculating \( r \) at angles like \( 0, \pi/2, \pi, \) and \( 2\pi \) can reveal significant patterns and curve behaviors.
• **Analyzing radial behavior**: Observing where \( r=0 \), particularly for limited theta ranges, helps identify curve loops or significant shapes.

Using these techniques allows for a better prediction of the curve's shape before fully sketching, making the graphing process more intuitive.

Radius and Angle

In polar coordinates, the radius \( r \) and angle \( \theta \) play crucial roles in defining the position of points on the curve. The radius \( r \) reflects the distance from the circle's center (pole) and varies as \( \theta \) changes. In limaçons with an inner loop, the radius becomes negative at certain angles (such as between \( \frac{\pi}{6} \) and \( \frac{11\pi}{6} \) in this exercise).
This behavior indicates that the points are being drawn in the opposite radial direction, forming an inner loop. Understanding these interactions between \( r \) and \( \theta \) is vital for accurately sketching polar curves and can lead to a deeper comprehension of more complex polar graphs.