Problem 28 - Precalculus with Limits

Question

In Exercises 15-28, identify the conic and sketch its graph. \(r=\dfrac{2}{2+3\sin\ \theta}\)

Step-by-Step Solution

Answer

The given polar equation represents a hyperbola. The graph could be sketched by calculating values of \(r\) for a variety of \(\theta\) values and plotting these points in a polar coordinate system.

Step 1: Identify the Conic Type

Observe the equation \(r=\dfrac{2}{2+3\sin\ \theta}\). Since \(\sin\ \theta\) is involved in the denominator of the equation and the coefficient of \(\sin\ \theta\) is positive, it means that the conic section in question is a conic with directrix parallel to the x-axis. If the coefficient had been negative, it would have been a conic with directrix parallel to the y-axis. This will lead us to conclude that this is a hyperbola.

Step 2: Sketch the Graph

The procedure for sketching the graph involves selecting a range of values for the angle \(\theta\) and then computing the corresponding values for \(r\) using the provided equation. Afterwards, mark these polar coordinates on a polar grid. As \(\theta\) varies from 0 to 2\(\pi\), plot each point and connect them to reveal the curve of the graph. Given the nature of computing these values, using a graphing calculator or software would be beneficial in accurately drawing the graph.

Key Concepts

Understanding Polar Coordinates

Polar coordinates are a two-dimensional coordinate system that offers an alternative to the Cartesian (x,y) coordinate system. In polar coordinates, each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point, known as the pole, is similar to the origin in the Cartesian system. The reference direction, typically the positive x-axis, is called the polar axis.

In mathematical terms, a polar coordinate \( (r, \theta) \) consists of:

• \(r\) - the radial distance from the pole, which can be thought of as how far away the point is from the center.
• \(\theta\) - the angle measured in radians or degrees, originating from the reference direction and moving counterclockwise.

Students often encounter polar coordinates when dealing with problems related to circular or spiral patterns. It's useful to visualize the radius \(r\) as a line segment from the pole to the point, and the angle \(\theta\) as the rotation from the reference line to the line segment.

Graphing a Hyperbola in Polar Coordinates

Graphing a hyperbola in polar coordinates can at first seem daunting, but understanding the equation's structure can provide a clear guide. A hyperbola in polar form can typically be recognized by its denominator which has a sine or cosine term.

To graph a hyperbola, such as \(r=\dfrac{2}{2+3\sin\ \theta}\), follow these simplified steps:

• Select values for \(\theta\) within the range [0, 2\pi].
• Calculate the corresponding \(r\)'s for each selected \(\theta\).
• Plot the points \((r, \theta)\) on a polar grid, connecting them smoothly to form the curve.

For our hyperbola, you'll notice the curves will have a symmetry with a slight distortion due to the \(\sin\ \theta\) term. When \(\theta\) equals 0 or \(\pi\), the \(r\) value will help to define the vertices of the hyperbola. It's recommended to use a graphing calculator or software to assist in sketching the graph for a more precise representation of the hyperbolic shape.

Conic Identification Techniques

Identifying conic sections from their polar equations requires a grasp of their different forms. In general, conics are determined by the presence of a denominator that contains either \(\sin\ \theta\) or \(\cos\ \theta\). The specific coefficients and constants within the equation can hint at which type of conic you are dealing with — ellipse, parabola, or hyperbola.

For the problem at hand, the equation \(r=\dfrac{2}{2+3\sin\ \theta}\) suggests a hyperbola because the \(\sin\ \theta\) coefficient is positive and in the denominator. Moreover,

• If the coefficient of \(\sin\ \theta\) or \(\cos\ \theta\) is positive, the corresponding directrix is parallel to the x-axis.
• If the coefficient is negative, the directrix is parallel to the y-axis.

Remember, the directrix of a conic section is a fixed line used in the formal definition of the shape. Hyperbolas will present two branches, one on each side of the center, showing a 'V' shape in Cartesian coordinates. Following these identification techniques makes recognizing conics simpler and more intuitive.