Problem 50 - Precalculus

Question

Graph equation. \(r=2+3 \sin \theta\)

Step-by-Step Solution

Answer

The graph is a limaçon with an inner loop due to \( b > a \).

Step 1: Understanding Polar Equations

The given equation is in polar coordinates, where the equation is expressed as a function of angle \( \theta \). In polar coordinates, each point is determined by an angle \( \theta \) and a radius \( r \). Here, \( r = 2 + 3 \sin \theta \), meaning the radius changes based on the sine of the angle.

Step 2: Identifying Type of Polar Graph

The equation \( r = 2 + 3 \sin \theta \) fits the form \( r = a + b \sin \theta \), which represents a limaçon. The limaçon can be a cardioid, inner loop, dimpled, or convex. Here, since \( b = 3 \) is greater than \( a = 2 \), the graph will have an inner loop.

Step 3: Determine Key Points

To graph the equation, calculate key points by varying \( \theta \) from \( 0 \) to \( 2\pi \). For example, at \( \theta = 0 \), \( r = 2 + 3 \sin 0 = 2 \); at \( \theta = \pi/2 \), \( r = 2 + 3 \cdot 1 = 5 \); at \( \theta = \pi \), \( r = 2 + 3 \sin \pi = 2 \); at \( \theta = 3\pi/2 \), \( r = 2 + 3 \cdot (-1) = -1 \).

Step 4: Graphing the Polar Coordinates

Plot these points on polar grid paper. Recall that a negative radius means the point is in the opposite direction through the origin. Draw smooth curves to connect these key points. The graph will have symmetry about the horizontal axis and a loop occurring in the bottom half of the graph due to the negative radius at \( \theta = 3\pi/2 \).

Step 5: Analyzing the Graph's Features

The graph is a limaçon with an inner loop. The maximum radius occurs at \( \theta = \pi/2 \) where \( r = 5 \). The inner loop occurs when \( r \) is negative, particularly at \( \theta = 3\pi/2 \). The graph should be symmetric about the horizontal axis, completing the loop from \( \theta \) moving from 0 to \( 2\pi \).

Key Concepts

Limaçon

A limaçon is a type of curve plotted in polar coordinates. It is identified by the equation form \( r = a + b \sin \theta \) or \( r = a + b \cos \theta \). Such curves belong to a family of polar plots that can exhibit different shapes based on the values of \( a \) and \( b \). In the context of the equation \( r = 2 + 3 \sin \theta \), we have a specific limaçon that will feature an inner loop. This is due to the condition that \( b > a \), meaning the inner loop appears when the sine part of the equation contributes a negative radius, specifically within some part of the cycle over \( \theta \) from \( 0 \) to \( 2\pi \).
The limaçon's unique appearance is defined by:

• Inner loop: Occurs when \( b > a \).
• Dimple or convex shape: Happens based on varying \( b \, \) (the greater \( b \) is, the loop is more prominent).
• Cardioid: A special case when \( a = b \).

Every variation of a limaçon provides unique insights into patterns and structures within polar graphs. Seeing these changes helps enhance the understanding of how equations translate into shapes.

Graphing Polar Equations

Graphing polar equations involves plotting points in a plane using polar coordinates with a radius \( r \) and an angle \( \theta \). The angle \( \theta \) is taken from the positive x-axis in a counter-clockwise direction, similar to how angles are measured in standard trigonometry. To graph an equation like \( r = 2 + 3 \sin \theta \), it's essential to compute key points.

• Calculate \( r \) for a range of \( \theta \): Start at \( \theta = 0 \) and progress to \( \theta = 2\pi \), capturing points where significant changes occur such as maximums and minimums.
• Plot on polar graph paper: Once these radian values and radii are noted, the points can be placed in the coordinate space that stems from the pole (or origin).
• Smooth curves: Join the plotted points with a smooth line to form the overall shape of the graph.

Often, polar graphs reveal intriguing patterns, such as loops or dimples, offering insights into the nature of the equation and its transformation of the circular function \( \theta \). Time spent plotting these accurately will pay off with clear visualization of the graph's features.

Radian Measure

Radian measure is a way of denoting angles using the radius of a circle. It is the standard measure used in calculus and advanced mathematics because it simplifies many mathematical expressions. Radian measure relates to angles as follows:

• One complete revolution around a circle: Equals \( 2\pi \) radians.
• Half a circle (straight angle): Is \( \pi \) radians.
• Quarter circle (right angle): Measures \( \frac{\pi}{2} \) radians.

When working in polar coordinates, angles like \( \theta = \pi/2 \) or \( \theta = 3\pi/2 \) are commonplace, and understanding how they translate to standard degrees can aid in predicting graph appearances. For example, \( \theta = \pi \) means rotating 180 degrees, while \( \theta = \frac{3\pi}{2} \) equates to a 270-degree rotation. Grasping radians as a unit provides better insight and fluency when manipulating angles within mathematical operations.

Symmetry in Polar Graphs

Understanding symmetry in polar graphs can simplify plotting and analysis greatly. Symmetry reduces computational effort and helps anticipate the shape's relationship with graph axes. There are three primary types of symmetry observed in polar graphs:

• Symmetry about the polar axis: This is the equivalent of symmetry about the x-axis in Cartesian coordinates. For any point \((r,\theta)\), there exists \((-r,-\theta)\).
• Symmetry about the line \( \theta = \frac{\pi}{2} \): Similar to y-axis symmetry; mirrored across this vertical line.
• Symmetry about the pole: Also called point symmetry, whereby for every point \((r,\theta)\), there exists \((r, \theta + \pi)\).

For the graph of \( r = 2 + 3 \sin \theta \), one can expect some symmetry primarily about the horizontal axis. Since \( \sin \theta \) is a symmetric function, the limaçon reflects a similar behavior over its cycle from \( 0 \) to \( 2\pi \). Recognizing these symmetries can dramatically streamline the process of sketching and understanding complex polar graphs.