Problem 44 - Precalculus with Calculus Previews

Question

Find a polar equation that has the same graph as the given rectangular equation. $$ x^{2}-12 y-36=0 $$

Step-by-Step Solution

Answer

The polar equation is \( r^2 \cos^2 \theta - 12r \sin \theta = 36 \).

Step 1: Convert the Rectangular Equation to Standard Form

The given rectangular equation is \( x^2 - 12y - 36 = 0 \). Let's rearrange this to express \( y \) in terms of \( x \): \( x^2 - 36 = 12y \). Hence, \( y = \frac{x^2 - 36}{12} \).

Step 2: Express in Polar Coordinates

In polar coordinates, \( x = r \cos \theta \) and \( y = r \sin \theta \). Replace \( x \) and \( y \) with these polar expressions in the equation derived in Step 1: \( r \sin \theta = \frac{(r \cos \theta)^2 - 36}{12} \).

Step 3: Simplify the Polar Equation

Replace \( (r \cos \theta)^2 \) with \( r^2 \cos^2 \theta \) to get \( r \sin \theta = \frac{r^2 \cos^2 \theta - 36}{12} \). Multiply both sides by 12 to clear the fraction: \( 12r \sin \theta = r^2 \cos^2 \theta - 36 \).

Step 4: Rearrange and Factor the Equation

Bring terms involving \( r \) on one side to get \( r^2 \cos^2 \theta - 12r \sin \theta = 36 \). Complete the expression to get a recognizable form, if possible.

Key Concepts

Rectangular Equation

The starting point for the problem is a rectangular equation, also known as a Cartesian equation. Rectangular equations involve variables like "x" and "y". These are connected with operations like addition, subtraction, and powers. Rectangular equations are based in the two-dimensional coordinate system where each point is described by an ordered pair

• (x, y)

The given rectangular equation is

• \(x^2 - 12y - 36 = 0\)

This equation represents a relationship between the x-coordinate and the y-coordinate. To convert this, we first rearrange it into an expression with "y" on one side, which makes it easier to interpret and manipulate in the following steps of graph conversions.

Graph Conversion

Graph conversion involves changing the form of an equation to suit a different coordinate system. In this case, we're converting from rectangular to polar coordinates. This step is crucial to better understand the nature of the graph that the equation represents. Polar coordinates use the distance from the origin and the angle from the positive x-axis.

• Begin by isolating the dependent variable, usually "y", in your rectangular equation.
• Then substitute the polar coordinate equivalents of "x" and "y", which are \( x = r \cos \theta \) and \( y = r \sin \theta \), into the rewritten equation.

By making these substitutions, you change the graph interpretation from rectangular to polar. This allows the same shape to be explored, but now with a focus on angles and radii instead of direct horizontal and vertical measures.

Polar Equation

A polar equation expresses the relationship between the radius "r" and the angle "θ" in a polar coordinate system. When dealing with problems requiring conversion to polar form, using known trigonometric identities allows us to replace rectangular terms with their polar counterparts effectively. The key relationship used here is:

• For x, substitute with \( r \cos \theta \)
• For y, substitute with \( r \sin \theta \)

For our exercise, the expression from the rectangular form becomes

• \( y = \frac{x^2 - 36}{12} \) which transforms into \( r \sin \theta = \frac{((r \cos \theta)^2) - 36}{12} \)

By applying trigonometry and simplifying, we achieve a recognizable polar form \( r^2 \cos^2 \theta - 12r \sin \theta = 36 \). This process showcases the power of symbolic manipulation in converting between graphing systems.

Trigonometric Functions

Trigonometric functions like sine and cosine link the sides of a right triangle to its angles. They are pivotal in transforming rectangular equations into polar forms because they bridge linear and angular measurements. Here's how they are applied:

• The cosine function, \( \cos \), relates the adjacent side over the hypotenuse in a right triangle. Here, it connects "x" with the radius "r". Hence, \( x = r \cos \theta \).
• The sine function, \( \sin \), represents the opposite side over the hypotenuse. It links "y" with "r", as in \( y = r \sin \theta \).

In polar equations, these functions are integrated to swap linear terms for angular expressions. Understanding these trigonometric roles can significantly ease the conversion and graph plotting processes, making trigonometry indispensable in coordinate transformations.