Problem 39 - Precalculus

Question

Sketch the graph of the polar equation using symmetry, zeros, maximum \(r\) -values, and any other additional points. $$r=5 \sin 2 \theta$$

Step-by-Step Solution

Answer

The graph of the polar equation \(r = 5 \sin 2\theta\) is a four-leaved rose with leaves facing North, South, East, and West, crossing the origin at angles 0, \(\pi/2\), \(\pi\), and \(3\pi/2\), and achieving its peak value of 5 at \(\pi/4\), \(3\pi/4\), \(5\pi/4\), and \(7\pi/4\).

Step 1: Determine the Symmetry

With polar equations, we essentially have three types of symmetry: symmetry about the x-axis, symmetry about the y-axis, and symmetry with respect to the origin. Test each type of symmetry by replacing \(\theta\) with \(-\theta\), \(\pi - \theta\), and \(\pi + \theta\) respectively. For the given equation \(r = 5 \sin 2\theta\), replacing \(\theta\) with \(-\theta\) gives \(r = 5 \sin 2(-\theta) = -5 \sin 2\theta = -r\). Thus the curve has symmetry with respect to the origin.

Step 2: Find the Zeros

The zero of \(r\) is the boundary line for the polar plot and occurs when \(\sin 2\theta = 0\). Solving this equation gives us \(\theta = 0, \pi/2, \pi, 3\pi/2\). This means that the curve will cross the origin at these values.

Step 3: Compute the Maximum \(r\)-Values

\[r\) reaches its maximum value when \(\sin 2\theta\) achieves its peak value of 1, which occurs at \(\theta = \pi/4, 3\pi/4, 5\pi/4, 7\pi/4\).

Step 4: Plot the Graph

Plot the curve starting by drawing boundary points at the angles determined in steps 2 and 3. Since this curve has symmetry with respect to the origin, you only need to plot it over a range of \(\pi\) radians and then generate the second half by reflecting the plot over the origin. You will obtain a four-leaved rose, with leaves facing the cardinal direction.

Key Concepts

Graphing Polar Equations

Polar equations, such as the given example \( r = 5 \sin 2 \theta \), represent curves using a specific coordinate system where each point on the plane is determined by an angle \( \theta \) and a radius \( r \). This is different from the typical Cartesian coordinates. In polar equations, \( r \) is the distance from the origin to a point, while \( \theta \) is the angle from the positive x-axis.

To graph such an equation, we plot points corresponding to various angle values \( \theta \). These points are determined by substituting chosen \( \theta \) values into the equation to get the corresponding \( r \)-values. By connecting these plotted points, a curve is formed which represents the equation in polar coordinates.

Graphing polar equations often results in interesting shapes, like circles, spirals, or roses, as in this case, where \( r = 5 \sin 2 \theta \) creates a four-leaved rose pattern.

Symmetry in Polar Graphs

Symmetry in polar graphs is a useful tool that simplifies the graphing process by reducing the need to plot unnecessary points. There are three tests of symmetry: symmetry about the x-axis, symmetry about the y-axis, and symmetry with respect to the origin.

To determine symmetry in the given equation \( r = 5 \sin 2 \theta \), we replace \( \theta \) with different values:

• \( -\theta \)
• \( \pi - \theta \)
• \( \pi + \theta \)

By substituting \( \theta \) with \( -\theta \), we find: \( r = 5 \sin 2(-\theta) = -5 \sin 2\theta = -r \), indicating that the graph is symmetric with respect to the origin.
Symmetry helps in predicting the other half of the graph without plotting additional points, thus making the graphing process more efficient.

Maximum Values in Polar Graphs

Maximum \( r \)-values are critical in plotting polar graphs as they determine the farthest points from the origin that the curve can reach. For the equation \( r = 5 \sin 2 \theta \), the maximum value of \( r \) occurs when \( \sin 2\theta = 1 \). This happens at specific angles:

• \( \theta = \pi/4 \)
• \( \theta = 3\pi/4 \)
• \( \theta = 5\pi/4 \)
• \( \theta = 7\pi/4 \)

At these angles, the equation gives maximum \( r \) values of 5. Plotting these points helps us outline the outer boundaries of the shape, which in this case, results in the tips of the "leaves" of a four-leaved rose.
The understanding of maximum values thus aids in accurately sketching the extent and the form of polar graphs.

Zeros of Polar Functions

Zeros of polar functions are points where the graph crosses the origin. These points occur when the \( r \)-value is zero, leading to an intersection at the origin. For the equation \( r = 5 \sin 2 \theta \), we determine zeros by setting \( \sin 2\theta = 0 \).

This condition is satisfied at the angles:

• \( \theta = 0 \)
• \( \theta = \pi/2 \)
• \( \theta = \pi \)
• \( \theta = 3\pi/2 \)

The graph touches the origin at these values of \( \theta \), indicating the "stems" or attachment points of the rose petals to the center of the polar graph.
Zeros are essential for understanding where the graph intersects itself or the origin, adding more precision to plotting polar curves.