Problem 50 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 45-78: Sketch the graph of the polar equation. $$ r=-2 \sin \theta $$

Step-by-Step Solution

Answer

The polar graph is a limaçon with an inner loop with points calculated for key angles to sketch the graph.

Step 1: Understand the Equation

The polar equation is given as \( r = -2 \sin \theta \). In polar coordinates, \( r \) is the radius (distance from the origin) and \( \theta \) is the angle measured from the positive x-axis.

Step 2: Identify the Type of Graph

The equation \( r = -2 \sin \theta \) represents a type of polar graph known as a limaçon, specifically with an inner loop, because the coefficient of \( \sin \theta \) is negative.

Step 3: Plot Key Angles

Evaluate \( r \) at key angles: - For \( \theta = 0 \), \( r = -2 \sin(0) = 0 \). - For \( \theta = \pi/2 \), \( r = -2 \sin(\pi/2) = -2 \). - For \( \theta = \pi \), \( r = -2 \sin(\pi) = 0 \). - For \( \theta = 3\pi/2 \), \( r = -2 \sin(3\pi/2) = 2 \).

Step 4: Determine Behavior in Each Quadrant

Analyze how \( r \) changes in each quadrant. - In the first quadrant (0 to \( \pi/2 \)), \( r \) becomes more negative as \( \theta \) increases.- In the second quadrant (\( \pi/2 \) to \( \pi \)), \( r \) decreases towards 0.- In the third quadrant (\( \pi \) to \( 3\pi/2 \)), \( r \) becomes positive.- In the fourth quadrant (\( 3\pi/2 \) to \( 2\pi \)), \( r \) returns towards 0.

Step 5: Sketch the Graph

Using the points and behavior identified, sketch the graph: - Begin at the pole (origin) at \( \theta = 0 \).- Draw the loop forming as \( r \) reaches -2 at \( \theta = \pi/2 \).- Return to the origin at \( \theta = \pi \).- The curve crosses through its maximum radius at \( \theta = 3\pi/2 \). - Complete the loop back to the origin as \( \theta \) approaches \( 2\pi \).

Key Concepts

Limaçon

A limaçon is a fascinating type of graph known in polar graphing for its unique shapes. The name "limaçon" comes from the French word for "snail" because the graph can resemble the shell of a snail.
There are different types of limaçons, including one with an inner loop, which is what we have in this exercise. This inner loop occurs when the equation of the limaçon has a negative coefficient in front of the sine or cosine function, such as in the equation \( r = -2 \sin \theta \).
The general form of a limaçon equation is \( r = a + b \cos \theta \) or \( r = a + b \sin \theta \). Depending on the values of \( a \) and \( b \), the graph can take different forms including a dimpled shape or a cardioid when \( a = b \).
The inner loop variation, like in our exercise, is specifically when \( |b| > |a| \), causing the graph to loop back on itself. Limaçons are quite captivating because they reveal how small changes in equations can drastically alter graph shapes.

Polar Coordinates

Polar coordinates offer a different way of representing points on a plane compared to the rectangular system we commonly use. In rectangular (Cartesian) coordinates, we think in terms of \( x \) and \( y \) coordinates, moving horizontally or vertically. However, in polar coordinates, points are determined by a radius and an angle.
In this system, any point is expressed as \( (r, \theta) \), where \( r \) is the radial distance from the origin (or pole) and \( \theta \) is the angle from the positive x-axis. This system is particularly useful for dealing with circular and spiral patterns, as it naturally accommodates rotation and radial distance.
Understanding plots in polar coordinates often involves imagining how you "sweep" around from the positive x-axis and how far outwards you go for the distance \( r \). Negative values for \( r \) mean we move in the opposite direction of the angle \( \theta \).

• Positive \( r \) values extend outward.
• Negative \( r \) values extend in the opposite direction from the angle.

Utilizing polar coordinates, especially in graphs like limaçons, highlights the elegance of curves that aren't so easily managed in the rectangular system.

Graphing Polar Equations

Graphing polar equations involves understanding and interpreting how equations that use polar coordinates translate into visual curves. To graph a polar equation like \( r = -2 \sin \theta \), first comprehend how changes in angle \( \theta \) affect the radius \( r \).
The process usually starts by calculating the radius for various key angles such as \( \theta = 0, \pi/2, \pi, 3\pi/2, \) and so on, which help map out the curve step-by-step. By plotting these coordinates in the polar plane and connecting them smoothly, you start to see the overall shape of the graph.
When sketching, consider:

• Symmetry: Some graphs are symmetric about the x or y axes or the origin itself.
• Behavior in Quadrants: Analyze how \( r \) changes as \( \theta \) moves through each quadrant.
• Special Points: Values where \( r = 0 \) indicate points where the curve crosses the pole.

The resulting graph for \( r = -2 \sin \theta \) shows an inner loop due to the negative coefficient of the sine function. Starting at the pole, the graph evolves by looping outwards and then returning to the origin, illustrating the complex yet stunning visuals of polar equation graphs.