Problem 6 - Precalculus

Question

Fill in the blanks. The equation \(r=1+\sin \theta\) represents a __________.

Step-by-Step Solution

Answer

The equation \(r=1+\sin \theta\) represents a cardioid.

Step 1: Identify the equation type

The equation is expressed in polar coordinates. There are a few types of graphs associated with straight forward polar equations: circles, limaçons, rose curves, and more. The equation \(r=1+\sin \theta\), lacking an additional theta term represents a limaçon.

Step 2: Recognize the specific type of limaçon

A limaçon may have an inner loop, a dimple, be convex, or be a cardioid. The particular specifics depend on the coefficients in the equation. In the polar equation \(r=1+\sin \theta\), the coefficient accompanying the sine function is 1, which is the same as the constant term. When these two are equal, this indicates that the limaçon is a cardioid.

Step 3: Fill in the blank

Given the analysis above, we can now confidently fill in the blank with the word 'cardioid'.

Key Concepts

Polar Coordinates

Polar coordinates provide a unique way of representing points in a plane. Unlike Cartesian coordinates, which use x and y to specify a location, polar coordinates utilize the distance from a reference point (called the pole) and an angle from a reference direction (often the positive x-axis). This system is especially useful for equations that describe curves emanating from a central point.
In polar coordinates, a point is denoted as \((r, \theta)\), where:

• \(r\) is the radius or distance from the pole.
• \(\theta\) is the angle in radians from the reference direction.

These coordinates are particularly beneficial in situations where symmetry and angles play a crucial role, like in circular and spiral patterns.

Limaçon

A limaçon is a type of curve that can be elegantly expressed in polar coordinates. The equation of a limaçon is typically in the form \(r = a + b \sin \theta\) or \(r = a + b \cos \theta\). Limaçons are characterized by the unique shapes they form which can include features like loops and dimples.
The specific shape of a limaçon is determined by the coefficients \(a\) and \(b\):

• If \(a = b\), the limaçon is a special type known as a cardioid.
• If \(a > b\), the limaçon is dimpled.
• If \(a < b\), the limaçon contains an inner loop.
• If \(a = 0\), it simplifies to a circle.

These curves are named after the French word limaçon, which means "snail," reflecting the spiral-like nature of some variations.

Sin Function

The sinusoidal function, often referred to simply as the sin function, maps an angle to a value between -1 and 1. It is periodic, repeating every 2\(\pi\) radians, making it ideal for modeling oscillations and waves.
In polar equations like \(r = 1 + \sin \theta\), the sin function modulates the radius based on the angle \(\theta\). This modulation affects the shape of polar curves, giving rise to intriguing patterns like those seen in limaçons.
The presence of the sin function introduces symmetry about the line \(\theta = \frac{\pi}{2}\) (the y-axis in polar terminology). This inherently affects the graph’s shape and is crucial in determining the specific type of limaçon when graphing.

Graph Types

When dealing with polar coordinate equations, there are several common graph types each having distinctive features:

• Circles: Simple symmetrical shapes, often centered at the pole, expressed by \(r = a\) or \(r = a \cos \theta\).
• Limaçons: These may have loops, dimples, or be convex, including the cardioid form as a special case.
• Rose Curves: Exhibiting petals, these curves are represented by \(r = a \sin(n\theta)\) or \(r = a \cos(n\theta)\).
• Spirals: Like the Archimedean spiral, represented by equations of the form \(r = a + b\theta\).

Each graph type has a standard polar equation that can be manipulated using coefficients to alter their shape and orientation, allowing for a wide range of visually striking curves.