Problem 14 - College Trigonometry

Question

Graph the following equations. $$ r=\frac{2}{1-2 \sin (\theta)} $$

Step-by-Step Solution

Answer

The graph is a hyperbola with its focus at the pole, symmetric about the y-axis, with a directrix at \( y = 0.5 \).

Step 1: Identify the Type of Polar Equation

The given equation is of the form \( r = \frac{ed}{1 - e\sin(\theta)} \), which is a conic section with the focus at the pole. In this case, \( e = 2 \) and \( d = 1 \). Since \( e > 1 \), the conic is a hyperbola.

Step 2: Find the Directrix of the Conic

The equation \( r = \frac{ed}{1 - e\sin(\theta)} \) can be rewritten as \( r = \frac{2 \cdot 1}{1 - 2\sin(\theta)} \). Comparing it with \( r = \frac{ed}{1 - e\sin(\theta)} \), since \( d = 1 \), the directrix is a horizontal line at \( y = 0.5 \) because for \( \sin(\theta) = \frac{1}{e} = \frac{1}{2} \), the term \( 1 - e\sin(\theta) \) vanishes.

Step 3: Analyze the Symmetry of the Graph

The equation involves \( \sin(\theta) \), which indicates symmetry about the vertical line through the pole, or the y-axis in Cartesian coordinates.

Step 4: Determine Key Angles and Points

Calculate values for specific angles, such as \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \), to determine key points. For example, at \( \theta = 0 \), \( r = \frac{2}{1} = 2 \); at \( \theta = \frac{\pi}{2} \), \( r = \frac{2}{1 - 2\cdot1} \), which is undefined (as expected for hyperbolic asymptote).

Step 5: Sketch the Graph Based on Calculations

Plot the pole, directrix, and calculated points on polar coordinates. Draw the hyperbola branches based on these, ensuring the graph adheres to the symmetries identified and approaches asymptotic directions derived from the undefined points found while calculating \( r \) values for angles.

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. They include circles, ellipses, parabolas, and hyperbolas. These shapes are significant in both mathematics and nature, as they describe the orbits of planets and the paths of projectiles. Each conic section can be defined by a specific type of equation. In polar coordinates, these equations look like

• Circle: \( r = a \)
• Ellipse: \( r = \frac{ed}{1 - e\cos(\theta)} \) with \( 0 < e < 1 \)
• Parabola: \( r = \frac{ed}{1 - e\cos(\theta)} \) with \( e = 1 \)
• Hyperbola: \( r = \frac{ed}{1 - e\cos(\theta)} \) with \( e > 1 \)

In these equations, \( e \) is known as the eccentricity, which dictates the shape and type of the conic. The directrix and focus are other essential parts to consider when dealing with polar equations. Each conic is symmetric in a certain manner, depending on its defining trigonometric function.

Hyperbola

A hyperbola is formed when the eccentricity \( e \) of a conic section is greater than 1. This particular configuration leads to an open curve with two separate branches, often resembling two mirrored arcs. The most recognizable property of a hyperbola is its asymptotic behavior, where the branches extend infinitely without ever meeting but always getting closer to an "asymptote." The standard polar form of a hyperbola is \[ r = \frac{ed}{1 - e\sin(\theta)} \] or \[ r = \frac{ed}{1 - e\cos(\theta)} \] depending on the orientation of the hyperbola.

• The focus is often positioned at the pole, playing a pivotal role in defining the hyperbola shape.
• Eccentricity greater than one dictates the hyperbola's openness.

Understanding how to convert these polar equations to their Cartesian equivalents can help visualize and graph the hyperbola more easily. Knowing that the hyperbola has two branches which never close gives insight into plotting its path.

Graphing Step-by-Step

Graphing polar equations like those that define conic sections requires a systematic approach: First, identify the equation type and ensure that all parameters align with it. For our given hyperbola, verify its form \[ r = \frac{2}{1-2\sin(\theta)} \] identifies it correctly, with \( e = 2 \).Once you've classified the conic,

• Locate the directrix, which generally complements the eccentricity and contributes positively to the graph's form.
• In this exercise, the directrix is detected at \( y = 0.5 \).

Next, analyze potential symmetry and calculate radial distances for different \( \theta \) values. Common angles such as \( \theta = 0, \frac{\pi}{2}, \pi, \) and \( \frac{3\pi}{2} \) are invaluable:

• Each angle provides critical intersection points or indicates asymptotic behavior when the equation is undefined.

Finally, sketch the graph.

• Mark the directrix and pole, and sketch the hyperbola branches while respecting the asymptotic implications.
• Checking the symmetry observed around axes can lead to a more precise graph.

Adhering to these planning steps ensures accurate rendering of these complex curves.