Problem 28 - Thomas Calculus

Question

Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian equations. Then describe or identify the graph. $$r \sin \theta=-1$$

Step-by-Step Solution

Answer

The Cartesian equation is \(y = -1\), a horizontal line at \(y = -1\).

Step 1: Identify the Polar Equation

Recognize the given polar equation to convert. Here, we have the equation: \[ r \sin \theta = -1 \]

Step 2: Recall Polar Coordinates to Cartesian Conversion

Recall that in polar coordinates, you can convert to Cartesian coordinates using the relationships: \[ x = r \cos \theta \] \[ y = r \sin \theta \]

Step 3: Substitute to Convert to Cartesian

Substitute the conversion formula for \(r \sin \theta\) which is \(y\), into our equation: \[ y = -1 \]

Step 4: Graph Description

Describe the graph corresponding to the equation \(y = -1\). This equation represents a horizontal line crossing the y-axis at -1.

Key Concepts

Polar Equations

Polar equations are mathematical expressions used to describe relationships in a plane with polar coordinates. In polar coordinates, each point in the plane is specified by an angle and a radius. The angle is usually denoted by \(\theta\) and the radius by \(r\). This coordinate system is very useful for equations involving curves that originate or return to a central point.
Polar equations can often look different from the Cartesian equations we may be used to. For instance, they often involve angle functions like sine or cosine. An example of a basic polar equation is \(r \sin \theta = -1\).
When working with polar equations, the first step is to recognize and understand the given polar equation. This understanding enables us to convert or transform the equation into other systems like Cartesian coordinates, which can be more intuitive.

• Polar equations are defined by radius \(r\) and angle \(\theta\).
• Used for systems with central points or rotational symmetry.
• Common in physics and engineering problems.

Cartesian Coordinates

Cartesian coordinates are a system that uses two or three numerical values to specify the position of a point on a plane or in space. Typically, the x-axis runs horizontally, and the y-axis runs vertically, with coordinates represented as \((x, y)\).
Conversion from polar to Cartesian coordinates involves using specific mathematical relationships. In the context of polar equations, we often convert them to Cartesian form to make graphing and interpretation easier.

• Use the relationships \(x = r \cos \theta\) and \(y = r \sin \theta\) for conversion.
• Polar equations \(r \sin \theta\) directly convert to \(y\) in Cartesian.
• Helps reveal line, circle, and other shape properties.

For the equation \(r \sin \theta = -1\), it converts to the simpler Cartesian equation \(y = -1\). This equation is a straightforward horizontal line on the Cartesian plane.

Graph Description

Graph descriptions help us visually understand mathematical equations and their solutions. For Cartesian equations like \(y = -1\), the graph represents a straightforward concept. This specific equation denotes a horizontal line on the Cartesian plane.
Here's what you should know about the graph of \(y = -1\):

• It's a horizontal line parallel to the x-axis.
• Crosses the y-axis at \(-1\), hence its name \(y = -1\).
• Stretches infinitely in the positive and negative x-direction.

Understanding the graph is crucial for visualizing how equations behave in a coordinate system. This enables clearer insights into their form and structure. Graphs can show us solutions, symmetry, and even intercepts with axes by translating the mathematical idea into visual stories.