Problem 78 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 45-78: Sketch the graph of the polar equation. $$ r=1-\csc \theta $$

Step-by-Step Solution

Answer

The curve is periodic with undefined points at \( \theta = n\pi \) and spirals, tending to negative infinity.

Step 1: Understand the Polar Equation

The given polar equation is \( r = 1 - \csc \theta \). In polar coordinates, \( r \) represents the radius, and \( \theta \) represents the angle. The function \( \csc \theta \) is the cosecant of \( \theta \), which is the reciprocal of \( \sin \theta \). Thus, \( \csc \theta = \frac{1}{\sin \theta} \).

Step 2: Analyze the Equation

Rewrite the equation using \( \csc \theta = \frac{1}{\sin \theta} \): \[ r = 1 - \frac{1}{\sin \theta} \]. The behavior of this equation depends on the value of \( \theta \), particularly where \( \sin \theta \) approaches zero, as \( \csc \theta \) will then approach infinity.

Step 3: Identify Key Points and Behavior

Examine the boundary conditions where \( \sin \theta \) is zero. This occurs at \( \theta = n\pi \), where \( n \) is an integer, resulting in \( r \to -\infty \). The function is undefined at these points. Analyze the regions in between, such as between \( \pi/2 \) and \( 3\pi/2 \), where \( \sin \theta \) is negative, yielding real negative \( r \) values.

Step 4: Sketch the Graph

Considering the previous steps, identify intervals for \( \theta \) and compute \( r \) values. For example, at \( \theta = \frac{\pi}{4} \), \( r = 1 - \sqrt{2} \). This polar equation leads to a curve that spirals around key points and is undefined at certain angles, resembling a periodic curve with sections tending to negative infinity.

Key Concepts

Cosecant Function

The cosecant function, represented as \( \csc \theta \), plays an important role in polar equations. It is the reciprocal of the sine function. Mathematically, this means \( \csc \theta = \frac{1}{\sin \theta} \). In simpler terms, whenever you see \( \csc \theta \), think about "how many times the sine of the angle fits into 1."
Understanding this can be crucial when dealing with polar equations, as the values of \( \csc \theta \) change significantly based on \( \theta \).
Here are some key points to consider:

• \( \csc \theta \) is undefined at points where \( \sin \theta = 0 \). This happens at multiples of \( \pi \) (e.g., \( 0, \pi, 2\pi, 3\pi, \) and so on).
• As \( \sin \theta \) gets closer to zero from a positive direction, \( \csc \theta \) grows towards positive infinity.
• When \( \sin \theta \) approaches zero from a negative angle, \( \csc \theta \) becomes more negative, heading towards negative infinity.
• \( \csc \theta \) has a periodic nature that can greatly affect the graph of polar equations.

Grasping these behaviors is essential to connect the abstract concept of cosecant with concrete graph sketching in polar coordinates.

Polar Equation Sketching

Sketching graphs in polar coordinates can be quite different from Cartesian plotting. Instead of \(x, y\) pairs, you work with \( r, \theta \) pairs. Here, \( r \) is the radius (or distance from the origin), and \( \theta \) is the angle measured from the positive x-axis.
Let's break down the process for sketching the given equation \( r = 1 - \csc \theta \):

• Identify the behavior of \( r \) based on different values of \( \theta \). Notice that whenever \( \theta \) approaches multiples of \( \pi \), \( \csc \theta \) becomes undefined, making \( r \) head towards negative infinity or undefined.
• Find values of \( \theta \) where \( \sin \theta \) is a significant number like \(0.5\) or \(1\). This helps in calculating and plotting corresponding \( r \) values.
• Consider the symmetry and periodic nature of \( \csc \theta \). This helps predict the repeating sections or shapes in the plot.

Using these steps can guide you to a smoother plotting experience. It aids in visualizing how the curve moves in a circular route, spiraling and forming sections that swap between negative and positive \( r \) values.

Behavior Analysis in Polar Coordinates

Understanding behavior in polar coordinates requires analyzing how \( r \) changes as \( \theta \) varies. In the equation \( r=1-\csc \theta \), we want to focus on key intervals and phenomena when \( \theta \) is varied.Analyzing the behavior involves:

• Identifying points where the function is undefined. These occur at \( \theta = n \pi \) (where \( n \) is an integer) as there, \( \sin \theta = 0\), hence \( \csc \theta \) is undefined and \( r \) moves towards negative infinity.
• Observing sections between undefined points. For instance, between \( \pi/2 \) and \( 3\pi/2 \), \( \sin \theta \) takes negative values. Thus, \( r = 1 - \frac{1}{\sin \theta} \) becomes positive or negative based on the sign of \( \sin \theta \).
• Recognizing periodicity in the pattern. Since any angle \( \theta \) in polar coordinates repeats every \( 2\pi \), expect the spiral-like plot of the function to reflect this repeating nature.

This thorough analysis not only helps in sketching accurate graphs but also forms a deeper knowledge of how polar equations behave differently compared to Cartesian equations. Through such analysis, one can better anticipate how these graphs will look, behaving like waves that undulate depending on \( \theta \) and \( \sin \theta \).