Problem 34 - Algebra and Trigonometry

Question

Sketch a graph of the polar equation. $$ r=2+\sin \theta $$

Step-by-Step Solution

Answer

The graph of \( r = 2 + \sin \theta \) is a limacon without an inner loop.

Step 1: Understanding Polar Equations

The given polar equation is of the form \( r = 2 + \sin \theta \), which describes a relationship in polar coordinates. Here, \( r \) represents the radial distance from the pole, and \( \theta \) is the angle measured from the positive x-axis.

Step 2: Identify Key Features

Recognize that the equation can form a limacon, a type of polar graph. The general form \( r = a + b\sin \theta \) with \( a = b = 2 \) suggests a limacon with an inner loop, as \( |b| > |a| \).

Step 3: Create a Table of Values

Calculate the values of \( r \) for various angles \( \theta \), such as \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2},\dots \). For \( \theta = 0 \), \( r = 2 + \sin(0) = 2 \). Continue for other typical angle values.

Step 4: Plot Points and Symmetry

Plot the calculated points on polar coordinates. For \( \theta = 0 \), plot (2, 0); for \( \theta = \frac{\pi}{2} \), plot (2.5, \frac{\pi}{2}); and so on. The graph is symmetric about the line \( \theta = \frac{\pi}{2} \).

Step 5: Identify the Inner Loop

Identify the inner loop by finding where \( r < 0 \). Solve for \( \theta \) when \( r = 0 \), we have \( 2 + \sin(\theta) = 0 \) leading to \( \sin(\theta) = -2 \) which is not possible, indicating no inner loop.

Step 6: Sketch the Graph

Using the plotted points and symmetry, complete the sketch. Draw curves connecting the points smoothly, characteristic of a limacon without an inner loop, starting from \( r = 1 \) and peaking at \( r = 3 \).

Key Concepts

Graphing Polar Coordinates

Graphing polar coordinates is a method to represent points on a plane using a radial distance and an angle. This differs from the traditional Cartesian system that uses x and y coordinates. In polar coordinates, each point is defined by:

• Radial distance (\( r \)): How far the point is from the origin or pole.
• Angle (\( \theta \)): The angle between the positive x-axis and the line from the pole to the point.

To plot a polar graph, select various angles for \( \theta \) and calculate the corresponding \( r \). For example, in the equation \( r = 2 + \sin \theta \), compute \( r \) for angles such as \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \), etc. Plot these polar coordinates on a graph and connect the points to see the complete shape. It's important to visualize these points by focusing on not just how far from the center they are, but in what direction they are located. By understanding these fundamentals of graphing polar coordinates, one can easily interpret and sketch the graph of any polar equation.

Limacon in Polar Graph

A limacon is a fascinating type of polar curve that can vary in shape. It can appear as a dimpled figure, resemble a heart, or form a loop depending on the parameters used in its equation. Limacons are generally expressed as \( r = a + b\sin \theta \) or \( r = a + b\cos \theta \). When graphing a limacon like \( r = 2 + \sin \theta \), we note the relation \( |b| > |a| \) does not apply here directly, meaning there is no inner loop but it could in other cases if these values were altered. The given equation forms a limacon without an inner loop, suggesting a distinct kind of symmetry and shape. Usually, one should look to see if \( a \) equals \( b \), as this indicates a common form of symmetry around the line \( \theta = \frac{\pi}{2} \), creating a smooth and uniform shape that many recognize as a hallmark of limacons.

Radial Distance and Angle in Polar Coordinates

Polar coordinates rely heavily on the concepts of radial distance and angle to define the position of a point. Radial distance (\( r \)) measures how far a point is from the origin, and the angle (\( \theta \)) indicates direction relative to the positive x-axis. In the context of our equation \( r = 2 + \sin \theta \), as \( \theta \) varies from \( 0 \) to \( 2\pi \), the distance \( r \) changes accordingly creating the limacon shape. Understanding these aspects allows us to determine exactly where each plot point lies. Key characteristics to note include:

• The direction of movement is counterclockwise as \( \theta \) increases.
• Negative \( r \) values represent points situated in a diametrically opposite direction from positive \( r \) values at the same \( \theta \).
• Symmetrical properties often help in sketching more complex shapes.

By grasping these foundational points, students can effectively map and sketch various polar equations.