Problem 35 - Thomas Calculus

Question

Replace the polar equations in Exercises \(23-48\) by equivalent Cartesian equations. Then describe or identify the graph. $$ r=\cot \theta \csc \theta $$

Step-by-Step Solution

Answer

The Cartesian equation is \(x = 0\), a vertical line at the y-axis.

Step 1: Express cotangent and cosecant in terms of sine and cosine

The given polar equation is \( r = \cot \theta \csc \theta \). Start by expressing \( \cot \theta \) and \( \csc \theta \) in terms of sine and cosine: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \).

Step 2: Simplify the expression

Substitute \( \cot \theta \) and \( \csc \theta \) into the equation: \[ r = \frac{\cos \theta}{\sin \theta} \cdot \frac{1}{\sin \theta} = \frac{\cos \theta}{\sin^2 \theta} \].

Step 3: Use polar to Cartesian conversion formulas

To convert to Cartesian coordinates, use the formulas: \( x = r \cos \theta \) and \( y = r \sin \theta \) where \( r = \sqrt{x^2 + y^2} \) and \( \tan \theta = \frac{y}{x} \). In addition, \( \cos \theta = \frac{x}{\sqrt{x^2 + y^2}} \) and \( \sin \theta = \frac{y}{\sqrt{x^2 + y^2}} \).

Step 4: Substitute polar coordinates with Cartesian equivalents

Substitute \( \cos \theta \) and \( \sin \theta \) with their Cartesian equivalents in \( r = \frac{\cos \theta}{\sin^2 \theta} \): \[ r = \frac{\frac{x}{\sqrt{x^2 + y^2}}}{\left(\frac{y}{\sqrt{x^2 + y^2}}\right)^2} = \frac{x}{\frac{y^2}{x^2 + y^2}} = \frac{x(x^2 + y^2)}{y^2} \].

Step 5: Multiply and simplify

Multiply the entire expression by \( y^2 \): \( r y^2 = x(x^2 + y^2) \). Recognizing \( r = \sqrt{x^2 + y^2} \), square both sides of \( r y^2 = x(x^2 + y^2) \) to eliminate \( r \): \[ (x^2 + y^2) y^4 = (x(x^2 + y^2))^2 \], which simplifies further to \( x^2 = 0 \). The equation \( x^2 = 0 \) indicates \( x = 0 \).

Step 6: Identify the graph

The equation \( x = 0 \) is the vertical line along the y-axis in the Cartesian plane.

Key Concepts

Cotangent in Polar Coordinates

When working with polar coordinates, it is important to understand how trigonometric functions like cotangent play a role in representing points on a plane. The cotangent of an angle, denoted as \( \cot \theta \), is the reciprocal of the tangent function. In terms of sine and cosine, it is expressed as:

• \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

In polar coordinates, the angle \( \theta \) is used along with the radius \( r \) to define the position of a point. The cotangent function can be understood as comparing the adjacent side to the opposite side of a right triangle formed by dropping a perpendicular from the point to the x-axis.
This relationship can be part of a transformation process when converting polar forms into Cartesian equations. Understanding cotangent in this context helps us easily navigate between coordinate systems and grasp how angles influence a point's position.

Cosecant in Polar Coordinates

Cosecant is another trigonometric function, closely connected to sine, often used in polar coordinates. Represented as \( \csc \theta \), the cosecant is the reciprocal of the sine function:

• \( \csc \theta = \frac{1}{\sin \theta} \)

This means that when \( \csc \theta \) is involved, it describes how far a point is from the origin with respect to the sine of the angle \( \theta \).
Working in polar coordinates, \( \csc \theta \) helps us relate the radius \( r \) to the vertical position of a point effectively. Through transformations and substitutions, as seen in steps for converting polar equations into Cartesian form, recognizing the role of cosecant is vital. It simplifies complex polar expressions by breaking them into more manageable components.

Graph Identification

Graph identification is crucial after transforming equations using trigonometric identities and coordinate conversions. The original exercise involves translating a polar equation into its Cartesian counterpart and understanding the graphical representation.
In the step-by-step solution, we see a sophisticated transformation concluding with \( x^2 = 0 \), leading to identifying the graph as a vertical line along the y-axis. This conclusion arises because:

• The equation \( x^2 = 0 \) simplifies to \( x = 0 \)
• \( x = 0 \) represents all points where the x-coordinate is zero

This denotes the y-axis itself, emphasizing how even equations seeming complex eventually describe simple geometric figures. Understanding the transformation sequence and identifying the graph ensures you can navigate various mathematical environments effectively, making you adept at interpreting different types of equations and their pictorial illustrations.