Problem 116 - Precalculus with Limits

Question

In Exercises 109-118, describe the graph of the polar equation and find the corresponding rectangular equation. Sketch its graph. \(r=-3\ \sin\ \theta\)

Step-by-Step Solution

Answer

The rectangular form of the equation from \(r = -3 \sin \theta\) is \(y = -3(1 - (x/\sqrt{x^2 + y^2})^2)\). The graph of this equation is a circle centered at the origin with a radius of 1.5 units. Because the r value is negative, the circle is below the x-axis.

Step 1: Convert Polar Equation to Rectangular Form

The conversion equations between polar coordinates (\(r, \theta\)) and rectangular coordinates (\(x, y\)) are \(x = r\cos \theta\), and \(y = r\sin \theta\). If we substitute \(r = -3\sin \theta\) into \(y = r \sin \theta\), we get \(y = -3\sin^2 \theta\). But, we know \(\sin^2 \theta = 1 - \cos^2 \theta\), so the rectangular equation becomes \(y = -3(1 - \cos^2 \theta)\). Substituting \(x = r\cos \theta\) into the equation gives \(y = -3(1 - (x/r)^2)\). Since \(r^2 = x^2 + y^2\), we can substitute \(r^2\) for \(x^2 + y^2\) and get \(y = -3(1 - (x/\sqrt{x^2 + y^2})^2)\). Simplifying this equation will give us the rectangular equation.

Step 2: Identify the Shape of the Graph

For \(r = a\sin \theta\), the graph has a shape of a circle with center at the origin and radius \(a/2\) if \(a > 0\) and radius \(-a/2\) if \(a < 0\). In case of \(r = -3\sin \theta\), since \(a = -3\), the graph is a circle with a radius of \(|-3/2| = 1.5\) units at the origin.

Step 3: Sketch the Graph of the Equation

Given that the circle's radius is 1.5 and is centered at the origin, you'll start by identifying the origin point (0,0). You'll then draw a circle around this point with a radius of 1.5 units. Note that since \(r = -3\sin \theta\), the circle is drawn in the negative direction, which means the graph will be below the x-axis.

Key Concepts

Polar Equations

Polar equations are mathematical expressions that define the relationship between the distance from the origin, denoted as r, and the angle from the positive x-axis, denoted as θ (theta), in polar coordinates. These coordinates are a two-dimensional coordinate system where each point on a plane is determined by a distance and an angle.

In our example, the polar equation is given by r = -3 sin θ. This defines a set of points for which the radius r varies with the angle θ. Understanding the behavior of the trigonometric function sin θ is key to analyzing the graph of this equation. Specifically, the negative sign in front of the 3 indicates a reflection across the polar axis (the line corresponding to θ = 0), which affects the direction in which we plot our graph.

Rectangular Equations

Rectangular equations, also known as Cartesian equations, define relationships between x and y coordinates on a typical two-dimensional graph. These equations are used in a Cartesian coordinate system, where each point is defined by an x-coordinate (horizontal positioning) and a y-coordinate (vertical positioning).

The conversion from polar to rectangular form involves the use of trigonometric identities and relationships between polar and rectangular coordinates. In the solution for our exercise, the polar equation r = -3 sin θ was converted into a rectangular form using the substitution method and trigonometric identities, ultimately transforming it to an equation solely in terms of x and y. This is particularly useful since rectangular equations are easier to graph on standard Cartesian grid paper.

Graph of Polar Equations

The graph of polar equations can often result in unique shapes that differ from those typically seen in rectangular coordinate systems. Due to the nature of polar equations involving angles and radii, the shapes may include circles, spirals, and other radial patterns that are not commonly found with rectangular equations.

In our problem, the graph of the polar equation r = -3 sin θ translates into a circle with specific characteristics. The negative coefficient of sin θ implies that the circle is located in the lower half of the polar coordinate system (below the polar axis). The step-by-step solution details that the radius of this circle is 1.5 units, based on the given equation. When sketching the graph, one would start at the origin and draw a circle with the determined radius, ensuring it reflects the correct positioning according to the sign and magnitude of the polar equation's coefficients.

Coordinate Systems

Coordinate systems are frameworks that enable the representation of points in space using a set of numbers, known as coordinates. There are various types of coordinate systems used in mathematics, but the most common are polar and rectangular (Cartesian) systems.

The polar coordinate system is based on angles and radii, while the rectangular system is based on perpendicular axis divisions denoted by x (horizontal axis) and y (vertical axis). Understanding both systems is crucial, especially when converting from one system to the other. A solid grasp of these systems allows us to interpret and graph equations in multiple ways, offering a broader perspective on problem-solving in mathematics.