Problem 57 - Precalculus with Calculus Previews

Question

Find a rectangular equation that has the same graph as the given polar equation. $$ r+5 \sin \theta=0 $$

Step-by-Step Solution

Answer

The rectangular equation is \( x^2 + y^2 + 5y = 0 \).

Step 1: Understand the Polar Equation

The given polar equation is \( r + 5 \sin \theta = 0 \). In polar coordinates, \( r \) represents the radius and \( \theta \) the angle from the positive x-axis.

Step 2: Rearrange the Polar Equation

Rearrange the given polar equation to express \( r \) in terms of \( \theta \): \( r = -5 \sin \theta \).

Step 3: Use the Polar to Rectangular Conversion

We use the polar to rectangular conversion formulas: \( x = r \cos \theta \) and \( y = r \sin \theta \). We also know that \( \sin \theta = \frac{y}{r} \).

Step 4: Substitute Sin Theta

Substitute the expression for \( \sin \theta \) into the equation for \( r \): \( r = -5 \times \frac{y}{r} \).

Step 5: Multiply to Clear the Fraction

Multiply both sides of the equation by \( r \) to eliminate the fraction: \( r^2 = -5y \).

Step 6: Express as a Rectangular Equation

Recall that \( r^2 = x^2 + y^2 \). Substitute \( r^2 \) in the equation: \( x^2 + y^2 = -5y \).

Step 7: Rearrange the Equation

Rearrange the equation \( x^2 + y^2 + 5y = 0 \) to get it in a standard form, if necessary.

Key Concepts

Polar Coordinates

Polar coordinates are a system of representing points in a plane using a distance and an angle. Unlike the rectangular coordinate system, which uses x and y coordinates to describe a point, polar coordinates use:

• \( r \): The radius, or the distance from the origin to the point
• \( \theta \): The angle from the positive x-axis to the point

This system is particularly useful in situations where circular or rotational symmetry is present. The conversion between polar coordinates and rectangular coordinates occurs through the formulas:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

These equations help translate back and forth between polar and rectangular systems, letting us analyze equations in the form naturally suited to the problem at hand. For example, the equation \( r = -5 \sin \theta \) describes a point location using the radius diminished by some factor related to the angle.

Rectangular Equation

To convert a polar equation into a rectangular one involves substituting polar elements with their corresponding rectangular equivalents. In our original exercise, the polar equation \( r + 5 \sin \theta = 0 \) is rearranged to \( r = -5 \sin \theta \). By substituting the relationships \( x = r \cos \theta \) and \( y = r \sin \theta \), we begin transforming the equation into a form expressing x and y only.
The key substitution usually centers around these compound trigonometric expressions:

• \( y = r \sin \theta \), giving \( \sin \theta = \frac{y}{r} \)
• Thus, when we place \( \sin \theta \) into \( r = -5 \sin \theta \), we obtain \( r = -5 \frac{y}{r} \)

Multiplying both sides by \( r \), results in the equation \( r^2 = -5y \). This is where the identity \( r^2 = x^2 + y^2 \) allows for the final transformation to the rectangular equation:

• \( x^2 + y^2 = -5y \)

This polynomial form represents the same graphically as the original polar equation.

Trigonometric Identities

Trigonometric identities help us convert and simplify between different types of equations. The key identities used in polar to rectangular conversion largely relate to sine and cosine functions, which bridge polar angles and rectangular distance:

• \( \sin \theta = \frac{y}{r} \)
• \( \cos \theta = \frac{x}{r} \)

For our exercise, using \( \sin \theta \) provided a straightforward path from polar to rectangular form. When we substitute the identity into the rearranged polar equation \( r = -5 \sin \theta \), it simplifies to \( r = -5 \frac{y}{r} \). Clearing fractions gives \( r^2 = -5y \), showcasing how identities facilitate simplification.
Finally, recognizing \( r^2 \) as \( x^2 + y^2 \) gives us a way to completely eliminate any \( r \) terms and conclude the conversion process. These identities are crucial tools in trigonometry, linking angles with linear and quadratic expressions used in algebra.