Problem 23 - Algebra and Trigonometry

Question

15–36 Sketch the graph of the polar equation. $$r=2-2 \cos \theta$$

Step-by-Step Solution

Answer

The graph of \( r = 2 - 2\cos\theta \) is a cardioid, symmetric about the polar axis.

Step 1: Identify the Type of Polar Equation

The given polar equation is \( r = 2 - 2 \cos \theta \), which is a specific form of a limacon. Limacons with the form \( r = a - a \cos \theta \) or \( r = a + a \cos \theta \) are known as cardiods. This is because the coefficients are equal, \( a = 2 \), indicating a cardiod shape.

Step 2: Determine the Symmetry

Since the equation involves \( \cos \theta \), it is symmetric about the polar axis (the horizontal axis in polar coordinates). Understanding this symmetry helps to sketch the graph more efficiently, as we can mirror our plot across this axis.

Step 3: Find Key Points

Let's find some key points to help sketch the graph: - When \( \theta = 0 \): \( r = 2 - 2\cos(0) = 0 \)- When \( \theta = \pi \): \( r = 2 - 2\cos(\pi) = 4 \)- When \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \): \( r = 2 - 2\cos(\frac{\pi}{2}) = 2 \)

Step 4: Plot Key Points

Using the polar coordinate system, plot the key points obtained in Step 3:- Point at \( \theta = 0 \) is the pole (0,0).- Point at \( \theta = \pi \) contributes a point at (4, \( \pi \)) which lies on the negative x-axis.- Points at \( \theta = \frac{\pi}{2} \) and \( \theta = \frac{3\pi}{2} \) result in a radius of 2, these are on the y-axis.

Step 5: Sketch the Graph Using Symmetry and Points

Now, use the identified symmetry (polar axis) and key points to sketch the graph. The graph should be heart-shaped (a cardioid), starting at the pole, peaking out to 4 units on the negative x-axis, circling around with these points guiding the outline of the cardioid.

Key Concepts

Limacon

In polar graphs, a limacon is a distinct mathematical curve that is often expressed in the form \( r = a + b \, \cos \theta \) or \( r = a + b \, \sin \theta \). This versatile polar curve presents itself in a variety of shapes depending on the values of \( a \) and \( b \). When sketching a limacon, pay attention to the relationship between \( a \) and \( b \):

• If \( |a| = |b| \), the limacon forms a special case known as a cardioid.
• If \( |a| > |b| \), it appears as a dimpled limacon with no inner loop.
• If \( |a| < |b| \), the limacon will feature an inner loop.

By understanding these properties, you'll be able to identify and sketch limacons more effectively in a polar coordinate setting.

Cardioid

A cardioid is a unique type of limacon, characterized by its heart-like shape. The cardioid occurs specifically when the coefficients in a polar equation are equal, such as in \( r = a + a \, \cos \theta \). In this form, the curve will always pass through the pole (origin) and peak outward. The symmetrical nature of the cardioid makes it easy to recognize:

• Its maximal radial distance from the pole is \( 2a \), either horizontally or vertically, based on if the cosine or sine is used.
• It has a cusp point at the pole where \( \theta = 0 \).
• It exhibits symmetry about the initial axis it aligns with.

With this type of curve, laying out initial plot points helps in accurately sketching the shape. Recognizing the cardioid quickly comes down to knowing these key features and the symmetrical peak that defines it.

Polar Coordinates

Polar coordinates are a two-dimensional coordinate system where every point in a plane is determined by distance from a reference point and angle from a reference direction. Unlike Cartesian coordinates which use x and y axes, polar coordinates express positions as \( (r, \theta) \), where:

• \( r \) is the radius or the distance from the origin (or pole).
• \( \theta \) is the angle measured in radians from the initial line or axis.

This system is incredibly beneficial when dealing with curves and paths that exhibit radial symmetry or are more naturally expressed in circular or spiral forms. For instance, curves like the limacon and cardioid are more succinctly represented in polar coordinates, which allows for a more intuitive sketching of otherwise complex paths.

Symmetry

Understanding symmetry in polar graphs helps in sketching and analyzing polar curves more efficiently. Symmetrical properties in polar equations provide clues about the shape and disposition of curves:

• Equations with \( \cos \theta \) are symmetric about the polar axis (horizontal axis).
• Equations involving \( \sin \theta \) are symmetric about the vertical axis, \( \theta = \frac{\pi}{2} \).
• If the polar equation appears unchanged when \( \theta \) is replaced with \( -\theta \), it indicates symmetry about the polar axis.
• Full symmetry often implies that half the graph can provide an entire picture by mirroring.

Identifying symmetry allows you to reduce plotting efforts, knowing you only need to calculate part of the curve firsthand. This tool is essential for efficiently sketching and understanding complex polar patterns like limacons and cardioids.