Problem 28 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 27-44: Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\)-plane. $$ r \sin \theta=-2 $$

Step-by-Step Solution

Answer

The Cartesian equation is \( y = -2 \). It's a horizontal line at \( y = -2 \).

Step 1: Convert Polar Equation to Cartesian Coordinates

The given polar equation is \( r \sin \theta = -2 \). We know that in polar coordinates, \( r \sin \theta = y \). Thus, the equation becomes \( y = -2 \) in Cartesian coordinates.

Step 2: Identify the Equation Type

The equation \( y = -2 \) represents a horizontal line in the Cartesian coordinate system that is 2 units below the x-axis.

Step 3: Graph in Cartesian Coordinates

To visualize this, draw a horizontal line that intersects the y-axis at \( y = -2 \). This line extends infinitely in the x-direction.

Step 4: Sketch the Graph in Polar Coordinates

In the polar coordinate system, the line \( r \sin \theta = -2 \) indicates that for any angle \( \theta \), the distance \( r \) causes the point to remain at a vertical position \( y = -2 \). It's a straight horizontal line at \( y = -2 \) on the Cartesian plane seen through polar perspective.

Key Concepts

Polar Coordinates

Polar coordinates are a way of describing the location of a point based on its distance from a central point, known as the origin, and the angle it makes with a reference direction, typically the positive x-axis. In the polar coordinate system, each point is represented as \(r, \theta\), where \(r\) is the radius or distance from the origin, and \(\theta\) is the angle in radians or degrees.

• The radius \(r\) can be positive or negative. A positive \(r\) indicates movement along the direction of the angle \(\theta\), while a negative \(r\) suggests movement in the opposite direction.
• The angle \(\theta\) is measured counter-clockwise from the positive x-axis.

Polar coordinates provide a powerful way to handle problems involving circular or rotational symmetry. This is particularly helpful in physics and engineering, where the geometry of problems often naturally lends itself to polar descriptions.

Cartesian Coordinates

Cartesian coordinates describe a point in a plane using two perpendicular axes that intersect at a point called the origin. These axes are typically labeled the x-axis (horizontal) and the y-axis (vertical). Each point in a Cartesian coordinate system is represented as \(x, y\), where \(x\) denotes the horizontal position and \(y\) denotes the vertical position.In contrast to polar coordinates:

• Coordinates are determined by a grid-like system, making it useful for linear measurement and straightforward calculation of distances and angles.
• In many contexts, Cartesian coordinates are often preferred for their simplicity and ease of visualizing linear relationships.

For example, the conversion of polar coordinates \(r \sin \theta = y\) to a Cartesian equation results in a straightforward horizontal line in this context, described simply by \(y = -2\). This reflects how different coordinate systems can provide unique insights into the same spatial relations.

Graphing Equations

Graphing equations involves plotting points on a coordinate plane to visually represent an equation. This visualization allows for understanding relationships between variables at a glance. In both polar and Cartesian systems, the goal is to present the underlying mathematical relationship in a form that is easy to interpret.For Cartesian coordinates:

• Equations like \(y = mx + b\) (a linear equation) draw straight lines, with \(m\) as the slope and \(b\) as the y-intercept.
• Graphing involves marking points that satisfy the equation and connecting them to reveal the shape of the graph.

In a polar coordinate system, graphing looks a bit different:

• Equations such as \(r = a \; or \; \theta = b\) determine curves based on radii and angles.
• This system is advantageous for capturing spirals and circles through polar equations, leveraging the natural circular symmetry they offer.

Finally, converting between these systems, as seen in converting \(r \sin \theta = -2\), is a crucial skill that links these geometric perspectives and expands our capacity to model and solve real-world problems visually.