Problem 28 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use symmetry to sketch the graph of the polar equation. Use a graphing utility to verify your graph. $$r=4(1+\sin \theta)$$

Step-by-Step Solution

Answer

First, understand the given polar equation represents a limacon with an inner loop. Recognize the vertical symmetry due to the \(\sin\) function and sketch key points. Reflect these in the y-axis. Finally, verify the accuracy of the sketch using a graphing utility.

Step 1: Understand the Polar Equation

The given polar equation is \(r = 4(1 + \sin(\theta))\). This is a form of a modified limacon (a polar curve) where the equation is \(r = a + b\sin(\theta)\) or \(r = a + b\cos(\theta)\). Here, \(a = 4\) and \(b = 4\), meaning this is a limacon with an inner loop because \(a = b\). There's a vertical symmetry line at \(\theta = \pi/2\) for the \(\sin\) function.

Step 2: Note the Symmetry and sketch

Since \(\sin(\theta)\) and \(\sin(-\theta)\) are equal in value, we have vertical symmetry around the y-axis, i.e., the line \(\theta = \pi/2\). To sketch, we can compute a few key points (where \(\sin(\theta)\) takes its minimum and maximum values -1 and 1, respectively) and reflect them in the y-axis.

Step 3: Use Graphing Tool for Verification

After obtaining the sketch, verify the graph using a graphing utility to ensure it's accurate. The tool will plot points for various values of \(\theta\) and for each, calculate the corresponding \(r\) using the polar equation. Compare this graph with the drawn sketch to verify.

Key Concepts

Polar Coordinates

The system of polar coordinates is a method of expressing locations in a plane using two numbers, which are typically denoted as \((r, \theta)\). Here, \(r\) represents the radial distance from the origin (or pole) to a point P, and \(\theta\) is the angle between the positive x-axis (or polar axis) and the line segment joining the origin to the point P.

Understanding polar coordinates is crucial for sketching polar equations. Unlike Cartesian coordinates, which use a grid of horizontal and vertical lines to specify points, polar coordinates rely on a grid of circles and radial lines. This system is particularly useful for dealing with curves that have symmetry around a central point or that follow a circular or spiral path.

Transitioning from Cartesian to Polar Coordinates

To convert from Cartesian coordinates \(x, y\) to polar coordinates \(\(r, \theta\)\), the following equations are used:

• \(r = \sqrt{x^2 + y^2}\)
• \(\theta = \arctan\left(\frac{y}{x}\right)\) (for \(x > 0\) and taking into account the correct quadrant)

When dealing with polar equations and their graphs, it's essential to think in terms of angles and distances from the origin rather than horizontal and vertical displacements.

Limacon Graph

A limacon graph represents a family of curves in the polar coordinate system characterized by the general equation \(r = a + b\sin(\theta)\) or \(r = a + b\cos(\theta)\). Here, the shape of the limacon depends on the relationship between the constants \(a\) and \(b\). The key variations include:

• If \(|a| > |b|\), the limacon does not have a loop.
• If \(|a| = |b|\), the limacon has an inner loop.
• If \(|a| < |b|\), the limacon has a dimple.

Sketching a Limacon

When sketching a limacon, it's important to identify these constants since they dictate the curve's form. For the equation \(r = 4(1 + \sin(\theta))\), it can be seen that \(a\) and \(b\) are both 4, thus creating a limacon with an inner loop.

To begin the sketching process, one should plot key points such as where \(\theta\) is 0, \(\frac{\pi}{2}\), \(\pi\), and \(\frac{3\pi}{2}\), corresponding to the cardinal directions. The points at which \(\sin(\theta)\) and \(\cos(\theta)\) reach their maximum and minimum (1 and -1) are also crucial for determining the overall shape of the graph.

Symmetry in Graphs

Exploring symmetry in polar graphs is a powerful way to simplify the sketching process, as it allows for drawing just a portion of the graph and reflecting it across a line of symmetry to complete the picture. Some common symmetries observed in polar graphs include:

• Line Symmetry: If a polar graph is symmetrical about a line (like the vertical line \(\theta = \pi/2\)), it means that for every point on the graph, there is another point directly opposite across the line of symmetry. This type of symmetry is often seen with graphs involving \(\sin(\theta)\) or \(\cos(\theta)\) functions.
• Polar Symmetry: Some graphs are symmetrical about the pole or origin. This means that if a point \((r, \theta)\) is on the graph, then so is the point \((r, \theta + \pi)\).
• Radial Symmetry: This occurs when a graph looks the same at any radial slice of the same angle, reflecting circular uniformity.

The equation \(r = 4(1 + \sin(\theta))\) exhibits line symmetry, as mentioned in the step-by-step solution. When trying to graph such an equation, one can cut down the work by plotting points for either \(\theta \geq 0\) or \(\theta < 0\) and then mirroring the plot across the appropriate line of symmetry, significantly simplifying the task at hand.