Problem 29 - Algebra and Trigonometry

Question

Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. $$ r \csc \theta=-2 $$

Step-by-Step Solution

Answer

The rectangular equation is \( y = -2 \). This is a horizontal line.

Step 1: Convert \( r \csc \theta = -2 \) to basic polar form

Recall that \( \csc \theta = \frac{1}{\sin \theta}\). Substitute this into the given equation:\[ r \frac{1}{\sin \theta} = -2 \]This simplifies to:\[ r = -2 \sin \theta \]

Step 2: Convert to rectangular coordinates

Use the polar to rectangular coordinate relationships: \( x = r \cos \theta \) and \( y = r \sin \theta \). Since from Step 1, we have \( r = -2 \sin \theta \), and recognizing \(r \sin \theta = y \), we can substitute directly:\[ y = -2 \]

Step 3: Identify the equation

The rectangular equation derived is \( y = -2 \). This is a horizontal line.

Step 4: Graph the equation

To graph \( y = -2 \), draw a horizontal line that passes through the y-axis at \( y = -2 \).

Key Concepts

Coordinate Transformation

One key concept in mathematics is learning how to transform coordinates, particularly from polar to rectangular coordinates. Polar coordinates are defined as \( (r, \theta) \), where \( r \) represents the radius (distance from the origin) and \( \theta \) represents the angle. Rectangular coordinates are defined as \( (x, y) \). To transform polar coordinates to rectangular coordinates, we use the relationships: \( x = r \cos \theta \) and \( y = r \sin \theta \). In this exercise, we convert the polar equation \( r \csc \theta = -2 \) to rectangular form. By substituting known trigonometric identities and simplifying, we transform it to \( y = -2 \). Understanding these transformations is crucial for solving many problems in physics, engineering, and navigation.

Polar Equations

Polar equations give us a different way of representing curves and shapes. They use the radius \( r \) and the angle \( \theta \) to describe points. Unlike rectangular equations, polar equations can make certain types of symmetry and curves easier to understand and work with. For example, circles, spirals, and rose curves are often simpler when described using polar equations. In the exercise, we start with \( r \csc \theta = -2 \). By using trigonometric relationships, we simplify this polar equation into a recognizable rectangular form with fewer steps: \( y = -2 \). Converting between these forms allows us to leverage the strengths of both systems for different types of problems.

Graphing Equations

Graphing equations is another fundamental skill. It helps us visualize what an equation represents. In rectangular coordinates, the equation \( y = -2 \) represents a horizontal line that crosses the y-axis at -2. To graph it, draw a straight line parallel to the x-axis and intersecting the y-axis at \( y = -2 \). By transforming and graphing equations, we not only understand their algebraic form but also their geometric representation, which is invaluable for solving real-world problems involving distances and angles.

Trigonometric Identities

One of the key tools in converting between coordinate systems is trigonometric identities. These identities provide relationships between different trigonometric functions. For instance, \( \csc \theta = \frac{1}{\sin \theta} \) is crucial for simplifying \( r \csc \theta = -2 \). Using this identity, we transform the equation to \( r = -2 \sin \theta \). Further, recognizing that \( r \sin \theta \) is simply \( y \) leads us directly to \( y = -2 \). Mastering these identities allows for quick and accurate transformations, enabling us to move fluidly between polar and rectangular forms.