Problem 24 - Algebra and Trigonometry

Question

15–36 Sketch the graph of the polar equation. $$r=1+\sin \theta$$

Step-by-Step Solution

Answer

The graph is a limaçon with an inner loop, having a maximum extent of 2 at \( \theta = \frac{\pi}{2} \).

Step 1: Identify the Polar Equation

We begin with the given polar equation \( r = 1 + \sin \theta \). This is a polar equation where \( r \) is expressed in terms of \( \theta \), using the sine function.

Step 2: Determine the Range of \( r \)

Since the sine function, \( \sin \theta \), varies from -1 to 1, the value of \( r = 1 + \sin \theta \) will range from 0 to 2 as \( \theta \) varies from 0 to \( 2\pi \).

Step 3: Sketch the Basic Shape

Because \( r = 1 + \sin \theta \), this is a limaçon with an inner loop. When \( \theta = \frac{3\pi}{2} \), \( r = 0 \), and when \( \theta = \frac{\pi}{2} \), \( r = 2 \). The graph will have a starting point at (0,0) when \( \theta = \frac{3\pi}{2} \) and loops out to its maximum extent of 2 units.

Step 4: Plot Key Points

Identify and plot key points such as where \( r = 0 \) (\( \theta = \frac{3\pi}{2} \)) and where \( r = 2 \) (\( \theta = \frac{\pi}{2} \)), along with other points like where \( \theta = 0 \) (\( r = 1 \)) and \( \theta = \pi \) (\( r = 1 \)).

Step 5: Draw the Limaçon

Connect the key points, firstly from the origin to the maximum \( r = 2 \) making a loop passing through all increments of \( \theta \), then looping back seamlessly to the origin denoting a smooth transition of values of \( r \) from 0 back to 2 and returning to 1.

Key Concepts

Limaçon

A limaçon is a special type of graph that can be plotted using polar coordinates. It is characterized by its unique shape which can have a dimple, loop, or be in the form of a cardioid (a heart-like shape). Limaçons are usually expressed in polar equations of the form \( r = a + b\sin(\theta) \) or \( r = a + b\cos(\theta) \). When exploring a limaçon, several outcomes are possible depending on the values of \( a \) and \( b \). For instance:

• If \( a < b \), the limaçon will have an inner loop, as in our example \( r = 1 + \sin(\theta) \).
• If \( a = b \), the graph forms a cardioid.
• If \( a > b \), the graph will not loop but may have a 'dimple' or be convex without a loop.

In our case, since \( a = 1 \) and \( b = 1 \), the resultant shape is a limaçon with an inner loop indicative of the graph's interesting characteristic transitions.

Polar Coordinates

Polar coordinates offer a unique way of plotting points, differing from the usual Cartesian coordinates. Here, each point on the plane is determined by a distance from a reference point (the pole or origin) and an angle from a reference direction (usually the positive x-axis).The coordinates are expressed as \( (r, \theta) \), where:

• \( r \) is the radius or the distance from the pole.
• \( \theta \) is the angle measured in radians from the positive x-axis.

To express and graph an equation in polar coordinates, focus on how the radius \( r \) changes with the angle \( \theta \). This dynamic is fundamentally different from Cartesian coordinates where you express \( y \) in terms of \( x \). For \( r = 1 + \sin(\theta) \), \( r \) varies smoothly from \( 0 \) to \( 2 \), creating an elegant looping path as \( \theta \) cycles from \( 0 \) to \( 2\pi \).

Trigonometric Functions

Trigonometric functions, such as sine and cosine, play a crucial role in creating polar graphs. These functions help describe the oscillations in \( r \) that produce curves like circles, spirals, and limaçons.For the equation \( r = 1 + \sin(\theta) \), the sine function dictates how the value of \( r \) changes as \( \theta \) changes:

• Sine (\

Graph Sketching

Graph sketching in polar coordinates requires a different approach than sketching graphs in the Cartesian system. In polar, you examine how \( r \) varies over a set of angles and plot corresponding points. Here are simplified steps to sketch a polar graph:

1. Determine basic characteristics like extrema by evaluating points at key angles (e.g., \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi \)) for \( r \).
2. Identify points where interesting features happen, such as an inner loop or a highest peak. For \( r = 1 + \sin(\theta) \), an inner loop occurs because \( r = 0 \) when \( \theta = \frac{3\pi}{2} \).
3. Plot these select points and smoothly connect them considering the nature of the trigonometric function involved.

The smooth and continuous increase and decrease of \( r \), due to the sine component, helps piece together the polar path as angles are traversed. This smoother and cyclical pattern is quintessential in sketching functions like limaçons.