Problem 9 - The Calculus with Analytic Geometry

Question

Draw a sketch of the graph of the given equation.\(r \sin \theta=-4\)

Step-by-Step Solution

Answer

The graph is a horizontal line at \( y = -4 \).

Step 1 Title - Understand the equation

Identify the form of the given polar equation. The equation is given as \( r \sin \theta = -4 \). This form suggests a relationship between the radius \(r\) and angle \( \theta \).

Step 2 Title - Rewrite using sine function

Recall that in polar coordinates, \( r \sin \theta \) represents the y-coordinate. Rewrite the equation to make it clearer: \( y = -4 \).

Step 3 Title - Recognize the type of figure

The equation \( y = -4 \) is a horizontal line in Cartesian coordinates where every point on the line has a y-coordinate of -4.

Step 4 Title - Sketch the graph

Draw a horizontal line at \( y = -4 \) on a Cartesian plane. Since \( y = r \sin \theta \) and \( r \) should be positive, indicate the points where the radius is positive.

Key Concepts

polar equation

In polar coordinates, equations are expressed in terms of the radius, \( r \), and the angle, \( \theta \). The given equation is \( r \sin \theta = -4 \), which may look different from familiar Cartesian equations. Here, \( r \) is the distance from the origin (or pole) to a point, and \( \theta \) is the angle formed with the positive x-axis (polar axis). To understand how this translates to a graph, it's helpful to connect polar coordinates to Cartesian coordinates, which use \( x \) and \( y \).

Remember that in polar coordinates:

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

This relationship helps in interpreting the polar equation. The term \( r \sin \theta \) simplifies to the y-coordinate in Cartesian coordinates. Therefore, the given equation effectively tells you that for any point on this graph, the y-coordinate must always be \( -4 \).

Cartesian coordinates

Cartesian coordinates are based on a rectangular coordinate system where each point can be defined by its distances along the x-axis and y-axis. The key takeaway from converting the polar coordinate to Cartesian is recognizing what the equation represents on the Cartesian plane.

Given the polar equation \( r \sin \theta = -4 \), when we recognize that this translates to \( y = -4 \) in Cartesian coordinates, it means the graph is a horizontal line across the entire x-axis at y = -4.

This horizontal line in the Cartesian coordinate system implies that for any x-value, the y-coordinate is fixed at -4. This clarity allows for straightforward graph sketching.

graph sketching

Graphing in polar coordinates can seem tricky, but it can be simplified by converting equations into Cartesian form when needed. In our case, we've translated the polar equation \( r \sin \theta = -4 \) to \( y = -4 \). Let's break down the steps for sketching this graph.

1. **Set up your Cartesian plane:** Draw the x and y axes on graph paper. Mark regular intervals along each axis to assist in plotting points accurately.

2. **Identify the equation:** In this example, \( y = -4 \) indicates a horizontal line.

3. **Draw the line:** Place your ruler horizontally at y = -4 and draw a straight line that spans the entire x-axis. This line represents all points such that the y-value is -4 for any x-value.

When sketching this on polar graph paper, keep in mind that the radius \( r \) will vary based on the angle \( \theta \). But the key point is that the vertical coordinate (essentially y) is always -4, just as shown in Cartesian coordinates. Make sure to indicate the appropriate lengths and segments on the Cartesian and polar graphs to maintain clarity for all viewers.