Question

Graph each polar equation. $$ r=\csc \theta-2, \quad 0<\theta<\pi \quad(\text { conchoid }) $$

Short answer

Graph the equation by calculating points for various \( \theta \) in the interval and connecting them, while addressing asymptotes and undefined points.

Step-by-step solution

Understand the polar equation

The given polar equation is \( r = \frac{1}{\text{sin}(\theta)} - 2 \). This describes how the radius \( r \) changes as the angle \( \theta \) changes, specifically in the range \( 0 < \theta < \frac{\beta}{\theta} < \frac{\beta}{2} = \text{\theta < \text{pi}} \).

Use the Cosecant Function Relationship

Recall that \( \text{csc}(\theta) = \frac{1}{\text{sin}(\theta)} \). Rewrite the equation as: \( r = \frac{1}{\text{sin}(\theta)} - 2 \).

Identify Critical Angles

Determine where \( \text{sin}(\theta) = 0 \) (undefined on the interval given). This will be important to understand the behaviour around specific points in our interval.

Plot Key Points

Calculate values of \( r \) for specific \( \theta \) values in the interval \( 0 < \theta < \frac{\beta < pi}} \). Example points for when \( \theta = \frac{\beta}{6, \theta = 6.7, and \theta = 0.245 } = 0.245 \)

Graph the equation

Plot the values of \( r \) corresponding to the \( \theta \) values found in the previous steps. Connect these points smoothly, considering the asymptotes and undefined points to graph the conchoid.

Review and Connect Points

Double-check the calculations and the graph to ensure a smooth continuous line connecting all the plotted critical points.

Key concepts

Cosecant Function

The given polar equation,
\[ r = \frac{1}{\text{sin}(\theta)} - 2 \]
uses the cosecant function.
The cosecant function, represented as \(\text{csc}(\theta)\), is the reciprocal of the sine function.
This means that \(\text{csc}(\theta) = \frac{1}{\text{sin}(\theta)}\).
Understanding this relationship helps us rewrite the original equation.
By rewriting it as
\[ r = \text{csc}(\theta) - 2 \]
we see directly how the radius \( r \) changes with the angle \( \theta \).
This is key for solving the problem because the cosecant function has specific properties that influence the graph.

• \(\text{csc}(\theta)\) is undefined when \(\text{sin}(\theta) = 0\), which occurs at integer multiples of \( \frac{\beta}{\theta} \), specifically at \(0\) and \(\pi\).
  
• \(\text{csc}(\theta)\) tends to infinity as \(\theta \) approaches these undefined points.
• The behavior of cosecant helps predict the nature of the graph, highlighting 'undefined points' and 'asymptotic behavior'.

Understanding the cosecant function is essential as it allows us to better graph polar coordinates involving it.

Graphing Polar Coordinates

Graphing polar coordinates involves plotting points based on their radius \( r \) and angle \( \theta \).
Polar graphs differ from Cartesian graphs since they represent points in a circular rather than a rectangular coordinate system.
For our equation,

• \( \text{r = \text{csc}(\theta)} -2 \) gets plotted by calculating \( r \) for different \( \theta \) values in the given range, \( 0 < \theta < \beta{\pi}\).
  
• Key angles we choose can be common values like \( \frac{\beta}{6}\), \( \frac{\beta}{3}\), \(\frac{\beta}{2}\), etc.
  These help in getting a rough idea of the graph's nature.

Consider these steps when graphing:

• Plot points by calculating \( r \) at chosen \( \theta \) intervals.
• Mark undefined points and asymptotes, where \( r \) tends towards infinity.
• Connect the plotted points smoothly, considering the behavior pattern (up-down, left-right) influenced by the sin/csc properties.

By following these steps, you can accurately graph any polar equation.

Critical Angles

Critical angles are specific \( \theta \) values where a function's behavior changes significantly.
For our equation, we identify critical angles by determining where the sine of \( \theta \) equals specific values.
Since \(\text{sin}(\theta) = 0\) at \(\theta = 0, \beta \), these points are essential because the cosecant function becomes undefined.

• Checking in the range \(0 < \theta < \beta \) reveals critical angles for the interval, as well as expected graph behavior.
• Other critical points include common values like \( \frac{\beta}{6}\), \( \frac{\beta}{3}\), and \( \frac{\beta}{2} \) where the sine function is known.
  

To best map out the graph, calculate \( r \) for these critical values:

• \( \theta = \frac{\beta}{6}\), \( \text{r} = \text{csc}(\theta) - 2\).
• \( \theta = \frac{\beta}{3}\), \( \text{r} = \text{csc}(\theta) - 2\).
• \( \theta = \text{\beta}{.245}\), \( \text{r} = \text{csc}(\theta) - 2\).

Identifying and using these critical angles gives us a better understanding of the function's key points and the overall shape of its graph.