Problem 39 - Thomas Calculus

Question

Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian equations. Then describe or identify the graph. $$r=\cot \theta \csc \theta$$

Step-by-Step Solution

Answer

The Cartesian equivalent is \( x^2 y^4 + y^6 = x^2 \). The graph is a complex curve.

Step 1: Start with the Given Polar Equation

The polar equation given is \( r = \cot \theta \csc \theta \). Our task is to convert this to a Cartesian form, traditionally given in \( x \) and \( y \).

Step 2: Express Trigonometric Functions in Cartesian Form

The trigonometric identities for \( \cot \theta \) and \( \csc \theta \) are \( \cot \theta = \frac{x}{y} \) and \( \csc \theta = \frac{1}{y} \). Substitute these into the original equation.

Step 3: Substitute and Simplify

Substituting these identities into the equation gives \( r = \frac{x}{y} \cdot \frac{1}{y} = \frac{x}{y^2} \). Since \( r = \sqrt{x^2 + y^2} \) in Cartesian coordinates, we substitute \( r \) to get \( \sqrt{x^2 + y^2} = \frac{x}{y^2} \).

Step 4: Square Both Sides

Square both sides of the equation to eliminate the square root. This yields \( x^2 + y^2 = \frac{x^2}{y^4} \).

Step 5: Clear the Fraction

Multiply every term by \( y^4 \) to get rid of the fraction: \( x^2 y^4 + y^6 = x^2 \).

Step 6: Simplify the Expression

Rearrange and simplify to find the Cartesian equation: \( x^2 y^4 + y^6 - x^2 = 0 \). This equation is now equivalent to the polar equation in Cartesian form.

Step 7: Identify the Graph

The Cartesian equation does not correspond to a standard graph like a circle or line, but describes a more complex curve. It represents a symmetrical shape intricately related to both \( x \) and powers of \( y \).

Key Concepts

Trigonometric Identities

In mathematics, trigonometric identities help us relate various trigonometric functions to each other and express them in different forms.
These identities are especially useful when we convert equations from polar to Cartesian form.When dealing with polar coordinates, we often encounter functions like \(\cot \theta\) and \(\csc \theta\).
These can be converted using well-known identities:

• \(\cot \theta = \frac{x}{y}\)
• \(\csc \theta = \frac{1}{y}\)

By substituting these into a polar equation, we can begin to express it in terms of \(x\) and \(y\).
This step is crucial as it lays the foundation for creating Cartesian equations from polar forms.
Ensuring you have a firm grasp on these identities will significantly ease the process of transformation between these coordinate systems.

Cartesian Equations

Once you switch a polar equation to Cartesian form, you deal with equations involving \(x\) and \(y\).
In the process of converting a polar equation like \(r = \cot \theta \csc \theta\), a series of algebraic manipulations yields a Cartesian equation.For instance, substituting trigonometric identities we previously discussed, you derive the equation:
\(\sqrt{x^2 + y^2} = \frac{x}{y^2}\).
Squaring both sides, and clearing fractions ultimately provides us the equation: \(x^2 y^4 + y^6 - x^2 = 0\).Working with Cartesian equations often involves simplifying expressions and handling algebraic terms efficiently.
Understanding these processes can be a bit challenging with complex equations, but practice makes it easier.Remember:

• Simplify the equation step-by-step.
• Square or manipulate terms as needed to obtain a clean expression.
• Clear fractions to simplify further steps.

These steps help us in connecting the geometric meaning from the polar form to a solvable Cartesian equation.

Graph Description

Understanding the graph of a Cartesian equation is like decoding the visual behavior of an algebraic expression.
The equation \(x^2 y^4 + y^6 - x^2 = 0\) doesn’t correspond to a simple geometric figure like a circle or line, but instead forms a more complex shape.Such equations describe curves that depict symmetrical properties or intricate relationships between \(x\) and different powers of \(y\).
Though these curves may not be immediately recognizable as standard figures, they still maintain specific characteristics defined by the equation.For students analyzing such graphs, it's helpful to:

• Look for symmetry around axes as potential clues.
• Examine-degree relationships in terms to guess curvature.
• Utilize graphing tools to visualize and better understand the equation’s representation.

Studying these characteristics helps in grasping how polar equations translate into geometrical shapes and behaviors in Cartesian systems.