Problem 66 - A Graphical Approach to Precalculus with Limits

Question

For each equation, find an equivalent equation in rectangular coordinates. Then graph the result. $$r=-5 \csc \theta$$

Step-by-Step Solution

Answer

The equivalent rectangular equation is \( y = -5 \). Graph it as a horizontal line at \( y = -5 \).

Step 1: Understand the Given Equation

The given polar equation is \( r = -5 \csc \theta \). Here, \( r \) represents the radius, and \( \csc \theta \) is the cosecant of the angle \( \theta \). Cosecant is the reciprocal of sine, so \( \csc \theta = \frac{1}{\sin \theta} \).

Step 2: Convert to Rectangular Coordinates

To convert the equation to rectangular coordinates, use the relationships: \( r = \sqrt{x^2 + y^2} \), \( \sin \theta = \frac{y}{r} \), and \( x = r \cos \theta \), \( y = r \sin \theta \). Substitute \( \csc \theta = \frac{1}{\sin \theta} = \frac{r}{y} \) into the given equation. Thus: \[ r = -5 \left( \frac{r}{y} \right) \].

Step 3: Simplify and Solve for \( r \)

Simplify the equation: \[ r = -5 \frac{r}{y} \]. Multiply both sides by \( y \): \[ ry = -5r \].

Step 4: Factor and Solve for y

Since the term \( r \) is common, divide by \( r \) (assuming \( r eq 0 \)): \[ y = -5 \]. This equation represents a horizontal line at \( y = -5 \) in rectangular coordinates.

Step 5: Graph the Rectangular Equation

Graphing the equation \( y = -5 \): draw a horizontal line on the Cartesian plane that passes through the point where \( y = -5 \). There is no dependence on \( x \), meaning this line is horizontal.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, often known as Cartesian coordinates, form a system where each point on a plane is determined by an ordered pair \((x, y)\). This system is named after René Descartes, who devised this method to describe geometrical shapes using algebra. In this coordinate system, "x" denotes the horizontal distance, while "y" represents the vertical distance from the origin.To visualize, imagine standing in a park. If you move "x" steps to the east and "y" steps north, you will arrive at a new location. This is how rectangular coordinates work. Each point is a combination of distances along the horizontal and vertical axes from the origin. In our problem, once converted from polar coordinates, the result was an equation that is already in simple rectangular terms, which is \(y = -5\). This signifies a horizontal line intersecting the "y" axis at -5.

Coordinate Conversion

Coordinate conversion refers to the process of transforming points from one coordinate system to another. For instance, to convert from polar to rectangular coordinates, you can utilize some key relationships between the two systems:

• \( r = \sqrt{x^2 + y^2} \)
• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

These relationships stem from the common properties of right triangles and trigonometric functions.In this exercise, the polar equation was \(r = -5 \csc \theta\). First, expressing \(\csc \theta\) as \(\frac{r}{y}\), we have \(r = -5 \frac{r}{y}\). Simplifying this equation and solving for \(y\) gave us \(y = -5\), which is a representation of a horizontal line in rectangular coordinates. This conversion allows us to graph the equation in a familiar coordinate plane format.

Graphing Equations

Graphing equations involves plotting a set of points that satisfy a given equation on a coordinate plane. In rectangular coordinates, this is particularly straightforward once we identify or simplify the equation.For our specific example, the equation \(y = -5\) is remarkably simple. It tells us that no matter what value "x" is, "y" will always be -5. This results in a horizontal line stretching infinitely along the "x" axis, parallel to it and intersecting the "y" axis at \(y = -5\).When graphing, you can start by plotting the y-intercept, which is -5 in this case, then draw the horizontal line across the graph. It's important to ensure that the line extends on both sides of the y-intercept, covering all possible values of "x". This visualization effectively represents the set of all points where "y" is consistently -5.