Problem 9 - Precalculus with Limits

Question

In Exercises 7-12, identify the type of polar graph. \(r=3(1-2 \cos \theta)\)

Step-by-Step Solution

Answer

The polar graph \(r = 3(1 - 2\cos\theta)\) is a convex limaçon oriented horizontally with the dimple on the left.

Step 1: Identify a and b

In the given equation \(r=3(1-2 \cos \theta)\), a is 3 (the coefficient of the bracket) and b is -2 (the coefficient of the cosθ term inside the bracket). It is important to know that the type of Limaçon depends on the absolute value of the ratio a/b.

Step 2: Identify type of Limaçon

The ratio a/b for this equation will be \(3/(-2)\) which equals -1.5. Absolute value of the ratio a/b is 1.5. The following are the conditions that determine the type of Limaçon: 1) If |a/b| < 1, it's a Limaçon with an inner loop. 2) If |a/b| = 1, it's a dimpled Limaçon. 3) If |a/b| > 1, it's a convex Limaçon, sometimes referred to as a Limaçon without a loop. Since |a/b| > 1, in this case, we have a convex Limaçon.

Step 3: Identify the Orientation

The orientation depends on the trigonometric function used in the equation. If cosθ is used, the Limaçon is horizontal with the loop or dimple on the right when the coefficient of cosθ is positive and to the left when it is negative. If sinθ is used, the Limaçon is vertical with the loop or dimple at the top when the coefficient of sinθ is positive and at the bottom when it is negative. The equation in the exercise includes a term -2 cos θ, so we know the Limaçon is a horizontal one with the loop or dimple on the left.

Key Concepts

Limaçon

A Limaçon is a type of polar graph known for its unique shapes and curves. These curves resemble a snail shell, which is where the name comes from, as "limaçon" is French for "snail". Limaçons are defined by equations in polar coordinates that take the form \(r = a + b \, \text{cos} \, \theta\) or \(r = a + b \, \text{sin} \, \theta\).

The shape of a Limaçon depends largely on the ratio \(\frac{a}{b}\). If the absolute value of \(\frac{a}{b}\) is less than 1, you get a Limaçon with an inner loop. If it equals 1, the Limaçon is dimpled, resembling a heart or an apple. When it is greater than 1, you're looking at a convex Limaçon, which appears more like a distorted circle without a loop.

Thus, determining \(\frac{a}{b}\) and its absolute value is essential in identifying what kind of Limaçon you are dealing with. In our problem, where \(r = 3(1-2 \cos \theta)\), \(\frac{3}{-2}\) has an absolute value of 1.5, indicating a convex Limaçon.

cosine function

The cosine function, denoted \(\cos \theta\), is a fundamental trigonometric function. It is used to explore relationships between angles in a right triangle and the lengths of the sides of that triangle, but it also extends to circular functions in a polar graph setup.

In the context of polar equations like \(r = 3(1-2 \cos \theta)\), the cosine function is essential in determining the orientation of the graph. The cosine function oscillates between -1 and 1, and this periodic behavior affects the distance \(r\) from the pole (the origin in polar coordinates) as \(\theta\) varies from 0 to \(2\pi\).

In polar graphing, when cosine is involved, the graph's overall shape takes on a specific directionality based on the value preceding \(\cos \theta\). In our example, \(-2 \cos \theta\) results in a horizontal Limaçon, with the concave part facing the left due to the negative coefficient.

polar coordinates

Polar coordinates are a way to describe the position of a point distinct from the traditional Cartesian coordinate system. Instead of using x and y-axis, polar coordinates rely on distance and angle, represented as \((r, \theta)\).

In this system, \(r\) represents the radial distance from the origin to a point, while \(\theta\) is the angle, measured in radians, from the positive x-axis to the point. This angle determines the direction from the origin.

Polar coordinates are especially useful for graphing curves that have symmetrical or circular properties, such as circles or Limaçons. The equations in polar form can produce curves such as spirals, roses, and of course, Limaçons, all of which can be more complex to represent with standard Cartesian coordinates.

graph orientation

Graph orientation in polar graphs refers to how a curve or shape is aligned within the coordinate system. Specifically, orientation involves both the direction of its features and its symmetry.

The trigonometric function utilized—\(\cos \theta\) or \(\sin \theta\)—has a major influence on orientation. For instance, equations with \(\cos \theta\) typically result in a graph that is oriented horizontally:

• A positive coefficient aligns a feature like a loop or dimple towards the right.
• Conversely, a negative coefficient directs it to the left.

For equations involving \(\sin \theta\), the graph tends to be vertical, with features oriented upwards or downwards depending on the coefficient’s sign.

In the exercise's equation, \(r = 3(1 - 2 \cos \theta)\), the negative coefficient of \(\cos \theta\) means the shape (a convex Limaçon) leans to the left on the polar plane.