Problem 39 - Precalculus with Limits

Question

In Exercises 23-48, 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 resulting graph of the polar equation \(r= 5\ sin\ 2\theta\) is two identical loops centered at the origin, reaching to 5 on either side. Careful plot of the zeros, maximum \(r\)-values, and connection of the plotted points gives the final shape of the graph.

Step 1: Determine the Symmetry

Analyze the function for symmetry. To do that, substitute \(-\theta\) for \(\theta\) and simplifying. If the equation remains the same, it has symmetry about the x-axis. The equation \(r = 5\ sin\ 2(-\theta) = -5\ sin\ 2\theta\) is not the same as the given equation, so the graph isn't symmetrical about the x-axis.

Step 2: Find the Zeros

Zeros of the function are values of \(\theta\) for which \(r = 0\). This happens when \(sin\ 2\theta = 0\). Solving this equation gives \(\theta = 0, \pi/2, \pi, 3\pi/2,...\). These are the points where the curve intersects the origin.

Step 3: Discover the Maximum \(r\)-values

Maximum \(r\)-values occur where the sine function reaches its maximum (1) or its minimum (-1). Therefore, the maximum \(r\)-values are 5 and -5.

Step 4: Plot Additional Crucial Points

Plot the points from steps 2 and 3 on the polar plane. The plot of these points will help in establishing the skirt of the polar equation.

Step 5: Connect the Plotted Points

Connect the plotted points in the order of increasing \(\theta\), from 0 to \(2\pi\). Considering the shape of the sine function and the points that have been plotted, draw a smooth curve that connects these points.

Key Concepts

Polar Coordinates

Polar coordinates provide an alternative to Cartesian coordinates for describing the positions of points in a plane. While Cartesian coordinates use a grid of vertical and horizontal lines, polar coordinates measure the distance and angle from a fixed point, known as the pole (analogous to the origin in Cartesian coordinates), and a fixed line, called the polar axis (similar to the x-axis in Cartesian).In polar coordinates, a point is represented by a pair \( (r, \theta) \), where \( r \) is the radius or the distance from the pole, and \( \theta \) is the angle from the polar axis. Positive \( r \) values mean the point is in the direction of the angle from the pole, while negative \( r \) values indicate the point is in the exact opposite direction. The angle \( \theta \) is typically measured in radians, where full rotation corresponds to \( 2\text{\pi} \) radians.

Symmetry in Polar Equations

Symmetry in polar equations is a property that can simplify the graphing process and provide insights into the shape of the graph. Common symmetries include symmetry with respect to the polar axis (\( \theta = 0 \) line), symmetry with respect to the line \( \theta = \text{\pi}/2 \) (similar to the y-axis in Cartesian coordinates), and symmetry with respect to the pole (origin).

Testing for Symmetry

• For symmetry about the polar axis, replace \( \theta \) with \( -\theta \) in the polar equation and see if the resulting equation is equivalent to the original. If so, the graph is symmetric about the polar axis.
• For symmetry about the line \( \theta = \text{\pi}/2 \) replace \( r \) with \( -r \) and \( \theta \) with \( \text{\pi} - \theta \) to check if the equation remains unchanged.
• To test for symmetry about the pole, replace \( r \) with \( -r \) and see if you get an equivalent equation. A graph with pole symmetry will look the same if flipped over the pole.

Zeros of Polar Functions

The zeros of a polar function are the angles \( \theta \) where the radius \( r \) becomes zero. These points indicate where the graph intersects the pole. Understanding where a polar function has its zeros helps to frame the graph and identify key points that the curve passes through.To find the zeros of the polar function \( r = 5 \text{\sin} 2\theta \), one sets the equation equal to zero and solves for \( \theta \). The zeros occur at angles where the \text{\sin} function is zero, which are integral multiples of \( \text{\pi} \) for \( 2\theta \). Thus, for \( r = 5 \text{\sin} 2\theta \), the zeros of \( r \) are obtained when \( \theta = 0, \text{\pi}/2, \text{\pi}, 3\text{\pi}/2, \text{\pi} \) and so on. These zeros are especially useful as starting points for plotting the graph.

Maximum r-values in Polar Graphs

The maximum \( r \) values in a polar graph are the furthest points from the pole along a particular direction. For the function \( r = 5 \text{\sin} 2\theta \), the maximum \( r \) values are where the sine function achieves its peaks, which occurs at angles where the sine is 1 or -1.

Finding Maximum \( r \) Values

• For positive peaks, set \( \text{\sin} 2\theta = 1 \) and solve for \( \theta \) to find the angles where \( r \) is maximum.
• For negative peaks, set \( \text{\sin} 2\theta = -1 \) and solve for \( \theta \) to determine the angles where \( r \) is at its minimum (which is the negative of the maximum value).

These maximum and minimum \( r \) values, along with the zeros of the function, help to sketch the extrema of the graph and capture its overall shape, ensuring a complete and accurate plot of the polar equation.