Problem 44 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 27-44: Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\)-plane. $$ r=6 \cot \theta $$

Step-by-Step Solution

Answer

The equivalent Cartesian equation is \( y^2(x^2 + y^2) = 36x^2 \).

Step 1: Recall relationship between Polar and Cartesian Coordinates

To express the polar equation in Cartesian coordinates, recall the conversions:\[ x = r \cos(\theta) \]\[ y = r \sin(\theta) \]\[ r^2 = x^2 + y^2 \]

Step 2: Express cotangent in terms of sine and cosine

The cotangent function can be expressed as the ratio of cosine to sine:\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \]

Step 3: Relate the given polar equation to Cartesian coordinates

Substitute the expression for \(\cot \theta\) into the given polar equation:\[ r = 6 \cdot \frac{\cos \theta}{\sin \theta} \]Multiply both sides by \(\sin \theta\) to isolate \(r \cos \theta\):\[ r \sin \theta = 6 \cos \theta \]So,\[ y = 6 \left( \frac{x}{r} \right) \]

Step 4: Substitute and simplify

Substitute \(y = r \sin \theta\) and \(x = r \cos \theta\) into the equation:\[ y = 6 \cdot \frac{x}{r} \]Multiply both sides by \(r\) to eliminate the fraction:\[ y r = 6x \]

Step 5: Express in terms of Cartesian form

Use the identity \(r^2 = x^2 + y^2\) to substitute for \(r\):\[ y \sqrt{x^2 + y^2} = 6x \]Squaring both sides to eliminate the square root:\[ y^2(x^2 + y^2) = 36x^2 \]This is the equation of the same graph expressed in Cartesian coordinates.

Key Concepts

Polar Coordinates

Polar coordinates are a system for representing points in a plane using a distance and an angle. Instead of defining a point by its distances along the horizontal and vertical axes like in Cartesian coordinates, polar coordinates use:

• The distance from a fixed point, often called the origin (represented as \( r \)).
• The angle \( \theta \) from a fixed direction, usually the positive x-axis.

This makes polar coordinates especially useful for problems involving circular or rotational symmetry. For instance, in the provided exercise, the polar equation is \( r = 6 \cot \theta \). Here, \( r \) varies based on the angle \( \theta \), generating the graph's shape around the origin.

Cartesian Coordinates

In contrast to polar coordinates, Cartesian coordinates define a point in a plane with two perpendicular values:

• The horizontal distance from the origin (\( x \)).
• The vertical distance from the origin (\( y \)).

Essentially, each point on the plane is defined by an \( (x, y) \) pair.
When converting from polar to Cartesian, formulas such as \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \) allow us to translate radial distances and angles into familiar axis distances.
Using these equations, we converted the given polar equation into Cartesian form, resulting in a relationship expressed with only \( x \) and \( y \).

Trigonometric Identities

Trigonometric identities are essential tools in angle and distance conversions. For this exercise, understanding how to rewrite trigonometric ratios matters. The identity for cotangent, \( \cot \theta = \frac{\cos \theta}{\sin \theta} \), is particularly useful here.
This identity allows rewriting parts of the polar equation as Cartesian terms, facilitating the transformation. This transformation makes simplifying and identifying suitable Cartesian expressions easier.
Knowing related identities, like the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \), also aids in verification and additional calculation adjustments.

Equation Conversion

Equation conversion is a process of transforming equations between coordinate systems. This involves linking polar and Cartesian forms to express a problem or graph in different ways. Our task required starting with \( r = 6 \cot \theta \) and deriving an equivalent Cartesian expression.
The steps included substituting trigonometric ratios and manipulating the equation based on known formulas. By multiplying and rearranging, we expressed everything in terms of \( x \) and \( y \):

• Started with polar equation and identified cotangent formula.
• Substituted and simplified using Cartesian substitution.
• Arrived at \( y^2(x^2 + y^2) = 36x^2 \), a Cartesian form.

Through this method, we constructed an equivalent expression that can now be graphed or analyzed in a Cartesian coordinate system.