Problem 42 - 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. \(r= 3\ \sin\ 3\theta\)

Step-by-Step Solution

Answer

The graph of \(r = 3 \sin 3\theta\) is a rose with 3 petals. The petals are at \(\theta = 0\), \(2\pi/3\) and \(4\pi/3\). The maximum r-value is 3.

Step 1: Identify the Pattern of Graph

Recognize that the function is a sine function that has been modified. It is in the form \(r = a \sin b\theta\), so it will have a general rose shape. The number of 'petals' in the rose will be determined by \(b\). In this case, \(b = 3\), so the graph will have 3 'petals'.

Step 2: Find the Zeros

Zeros of the function are the points where \(r = 0\). Given the function is \(r = 3 \sin 3\theta\), setting \(r\) equal to zero gives us \(\sin 3\theta = 0\). Using the property of sine function, that \(\sin n\pi = 0\), where \(n\) is an integer, we find that \(\theta = n\pi/3\) are zeros.

Step 3: Find the Maximum r-values

The maximum \(r\)-values occur at the maximum points of the sine function. We know that the maximum value of \(\sin x\) is 1. Therefore, when \(\sin 3\theta = 1\), \(r\) will be at its maximum, which is 3.

Step 4: Sketch the Graph

Start by drawing a proper polar coordinate system. Plot the spot where \(\theta = 0\), which will act as one of the 'petals' of the rose. Also mark the spot where \(\theta = 2\pi/3\) and \(\theta = 4\pi/3\). All three points are at a distance \(r = 3\) units from the origin. Also, mark the zeros of \(\theta\). These points provide a skeleton to draw 3 'petals' of the rose.

Key Concepts

Polar Equation

A polar equation is an expression that defines a relationship between the radius "\(r\)" and the angle "\(\theta\)" in a polar coordinate system. Unlike the Cartesian plane which uses \(x\) (horizontal) and \(y\) (vertical) axes, polar coordinates focus on the distance from the origin and the angle from a fixed direction, usually the positive \(x\)-axis.
In polar equations, you often see forms like \(r = \text{{some function of }} \theta\), which dictates how the graph behaves as \(\theta\) changes.
For example, in our equation \(r = 3 \sin 3\theta\), the expression is bounded by the trigonometric sine function. This implies a periodic and symmetrical graph due to the repetitive nature of the sine function. In this context, varying \(\theta\) results in a dynamic change in the distance \(r\) from the origin, creating unique and interesting shapes like spirals and roses.

Graph Sketching

Graph sketching in polar coordinates involves plotting points based on the relationship between \(r\) and \(\theta\), and then connecting these points to form the complete graph. Here are some steps to use when sketching:

• Identify symmetry: Determine if the graph has symmetry with respect to the polar axis, line \(\theta = \pi/2\), or the pole (origin).
• Locate Zeros: Find where the radius \(r\) equals zero, which usually indicates the graph will intersect the pole.
• Determine Maximums: Identify angles where \(r\) is at a maximum to quantify the "reach" of the graph.
• Plot Additional Points: For better accuracy, select other angles for \(\theta\) to identify additional points on the graph.

For the polar equation \(r = 3 \sin 3\theta\), these steps help in visualizing a three-petaled structure called a "rose curve" by defining where the petals start, end, and how far they extend from the origin.

Sine Function

The sine function is a foundational trigonometric function that oscillates between -1 and 1. It is periodic, with each cycle repeating every \(2\pi\) radians.
In polar equations like \(r = 3 \sin 3\theta\), the sine function directly influences the radius \(r\) as \(\theta\) increases. The amplitude of the sine function affects the maximum value of \(r\), while the coefficient of \(\theta\) (in this case, 3) alters the frequency or number of cycles within a \(2\pi\) interval.
Thus, \(r = 3 \sin 3\theta\) demonstrates how the sine wave's properties can create complex patterns in polar coordinates, leading to elegant graphical representations such as the "rose" structure observed in this exercise.

Zeros of Polar Function

Zeros of a polar function are the angles \(\theta\) where the radius \(r\) becomes zero. These points are crucial because they typically represent intersections of the graph with the pole (origin).
To find the zeros in \(r = 3 \sin 3\theta\), set the function equal to zero: \(3 \sin 3\theta = 0\). Solving \(\sin 3\theta = 0\) gives \(3\theta = n\pi\), where \(n\) is an integer. Thus, \(\theta = n\pi/3\) are the zeros.
These zeros form the basis for plotting the polar graph because they dictate critical points where the graph shifts direction or returns to the pole. Identifying zeros gives a framework within which other features of the curve, such as loops or petals, can be accurately sketched on the polar plot.