Problem 42 - Calculus Early Transcendentals

Question

\(29-48\) Sketch the curve with the given polar equation. $$r=2+\sin \theta$$

Step-by-Step Solution

Answer

The sketch is a limaçon with a dimple, symmetric about the vertical axis.

Step 1: Identify the Given Polar Equation

We have the polar equation \(r = 2 + \sin \theta\). This equation represents a curve in polar coordinates. Here, \(r\) is the radius or distance from the origin, and \(\theta\) is the angle.

Step 2: Determine the Range of \(r\)

Since \(\sin \theta\) varies between \(-1\) and \(1\), the radius \(r\) in this equation, \(r = 2 + \sin \theta\), will vary from \(1\) to \(3\).

Step 3: Analyze Critical Points

When \(\theta = \frac{\pi}{2}\), \(\sin \theta = 1\) and \(r = 3\). When \(\theta = \frac{3\pi}{2}\), \(\sin \theta = -1\) and \(r = 1\). These are the maximum and minimum values for \(r\), respectively.

Step 4: Sketch Points for Specific Angles

For key angles, calculate the corresponding \(r\) values. For example, when \(\theta = 0\), \(\sin \theta = 0\), so \(r = 2\). For \(\theta = \pi\), \(\sin \theta = 0\), so \(r = 2\). Plot these points on a polar grid.

Step 5: Draw the Curve

Using the points plotted and the variation of \(r\) from \(1\) to \(3\), sketch the curve. The maximum radial distance occurs at \(\theta = \frac{\pi}{2}\) forming the outermost point of a loop, and the minimum distance occurs at \(\theta = \frac{3\pi}{2}\), creating the inner cusp.

Step 6: Label the Polar Curve

The resulting sketch is a limaçon with an inner dimple. It is symmetric about the line \(\theta = \frac{\pi}{2}\) due to the nature of the function \(\sin \theta\).

Key Concepts

Understanding Polar Coordinates

Polar coordinates are a way of representing points in a plane using a radius and angle. Instead of using the Cartesian coordinates \((x, y)\), polar coordinates describe a point by how far it is from the origin, called the radius \(r\), and the angle \(\theta\) made with the positive x-axis. This system can be especially useful for dealing with problems involving rotations and circular motion.

Some of the key aspects of polar coordinates include:

• The radius \(r\) indicates the distance of the point from the origin. A positive value of \(r\) means the distance in the direction of the angle \(\theta\).
• The angle \(\theta\) specifies the direction of the radius, measured counterclockwise from the positive x-axis.
• Polar equations can often be more straightforward than Cartesian equations for curves like spirals or circles.

Understanding the range of \(\theta\) and how it impacts the value of \(r\) is crucial in sketching and interpreting polar graphs.

Exploring the Limaçon Shape

A limaçon is a type of polar curve which can produce an array of visually interesting shapes, depending on the parameters in its defining equation. In our example, the polar equation \(r = 2 + \sin \theta\) creates a limaçon with an inner dimple.

Limaçons are defined commonly as a form \(r = a + b\sin\theta\) or \(r = a + b\cos\theta\):

• If \(a > b\), the limaçon has an inner dimple but does not pass through the origin.
• If \(a = b\), the limaçon passes through the origin, producing a cardioid shape.
• If \(a < b\), the limaçon forms a loop.

This classification helps in predicting the shape when graphing. In the equation \(r = 2 + \sin \theta\), we have \(a = 2\) and \(b = 1\), so \(a > b\), producing a dimpled limaçon.

The Role of Trigonometric Functions

Trigonometric functions like \(\sin\theta\) and \(\cos\theta\) are integral to polar equations because they define how \(r\) varies with \(\theta\). In the equation \(r = 2 + \sin \theta\), \(\sin\theta\) influences the variation in \(r\) as \(\theta\) changes.

Key points to consider when dealing with trigonometric functions in polar curves include:

• The function \(\sin \theta\) oscillates between \(-1\) and \(1\), giving the possible range of \(r\) in this scenario from \(1\) (when \(\sin \theta = -1\)) to \(3\) (when \(\sin \theta = 1\)).
• Characteristic angles, like 0, \(\frac{\pi}{2}\), \(\pi\), and \(\frac{3\pi}{2}\), help in sketching by providing specific \(r\) values which denote points on the curve.
• The symmetry of trigonometric functions can also assist in predicting and drawing graphs, as these functions often result in symmetric polar curves.

Understanding these behaviors allows us to predict not only the shape of our curve but also how it is positioned and drawn on the polar grid.