Question

Equations of the form \(r=a \sin m \theta\) or \(r=a \cos m \theta,\) where \(a\) and \(b\) are real numbers and \(m\) is a positive integer, have graphs known as roses (see Example 6 ). Graph the following roses. $$r=6 \sin 5 \theta$$

Short answer

Answer: The rose graph has 5 petals, and they are oriented in the vertical direction, with the first petal pointing upwards.

Step-by-step solution

Determine the number of petals

The number of petals is determined by the value of \(m\). In this case, \(m = 5\). Since it is an odd number, the rose will have \(5\) petals.

Determine the length of the petals

The length of the petals depends on the coefficient \(a\). In this case, \(a = 6\). Thus, the length of the petals is \(6\).

Determine the orientation of the petals

Since this equation uses \(\sin\) instead of \(\cos\), the rose petals will be oriented in the vertical direction, with the first petal pointing upwards.

Plot points for each petal and connect them to create the graph

To graph the rose, let's start by selecting points along each petal. Using the equation \(r = 6 \sin 5 \theta\), we can find the points for each petal by varying \(\theta\) in small increments. For example, using \(\theta\) increments of \(\frac{\pi}{30}\), we can find points along the first petal by plugging in values for \(\theta\), such as \(\frac{\pi}{30}, \frac{2\pi}{30},\dots , \frac{18\pi}{30}\) into the equation. We can follow the same process to find points for the other petals. After finding a sufficient number of points for each petal, plot them on the polar coordinate plane and connect them to create the graph of the rose. Here are the main points for each petal: - Petal 1: \(\frac{\pi}{30}, \frac{2\pi}{30}, \dots, \frac{18\pi}{30}\) - Petal 2: \(\frac{19\pi}{30}, \frac{20\pi}{30}, \dots, \frac{36\pi}{30}\) - Petal 3: \(\frac{37\pi}{30}, \frac{38\pi}{30}, \dots, \frac{54\pi}{30}\) - Petal 4: \(\frac{55\pi}{30}, \frac{56\pi}{30}, \dots, \frac{72\pi}{30}\) - Petal 5: \(\frac{73\pi}{30}, \frac{74\pi}{30}, \dots, \frac{90\pi}{30}\) Now you can graph the rose \(r = 6 \sin 5 \theta\) by connecting these points on the polar coordinate plane.

Key concepts

Rose Curves

Rose curves are a fascinating family of curves that emerge in polar coordinates, which are defined by equations like \(r = a \sin m \theta\) or \(r = a \cos m \theta\). These curves are known as roses due to their petal-like shapes.

• Number of Petals: The number of petals of a rose curve is determined by the value of \(m\). If \(m\) is odd, the rose has \(m\) petals. If \(m\) is even, the rose has \(2m\) petals.
• Length of Petals: The length of each petal is given by the absolute value of \(a\). This dictates how far each petal reaches from the origin.
• Symmetry and Orientation: The function \(\sin\) and \(\cos\) determine the orientation. A rose curve using \(\sin\) tends to have its first petal oriented vertically, whereas \(\cos\) orients it horizontally.

Understanding these characteristics allows us to identify the structure and properties of rose curves before even plotting them.

In the given example, \(r = 6 \sin 5 \theta\), we have a rose with 5 petals, each extending 6 units from the center. The use of the sine function implies a vertical orientation for the first petal.

Graphing Polar Equations

Graphing polar equations might seem daunting at first, but it becomes straightforward once the fundamental concept is grasped. Unlike Cartesian coordinates that use \(x\) and \(y\), polar coordinates utilize \(r\) (radius) and \(\theta\) (angle).

• Plotting Process: To plot a point in polar coordinates, determine its radial distance (\(r\)) from the origin and the angle (\(\theta\)) it makes with the positive x-axis.
• Incremental Steps: When graphing equations like rose curves, it’s effective to choose consistent increments for \(\theta\). This helps in generating enough points to outline the shape smoothly.
• Connecting Points: After plotting points, connect them with smooth curves to visualize the rose or any given polar graph. Careful plotting ensures accuracy in the representation.

For \(r = 6 \sin 5 \theta\), plotting becomes organized by selecting \(\theta\) values systematically and calculating corresponding \(r\) values, resulting in the rose curve structure.

Trigonometric Functions

Trigonometric functions are the bedrock of polar equations, like those generating rose curves. They bring periodicity and symmetry to the graphs.

• Sine Function: Yields curves starting and ending at the origin with symmetry about \(\theta = \frac{\pi}{2}\). The maxima occur at \(\frac{\pi}{2m}\) intervals if \(m\) petals are present.
• Cosine Function: Similar to sine but symmetric about the polar axis (horizontal). The first maximum of cosine-oriented roses typically appears along the positive x-axis.
• Amplitude and Frequency: The coefficient \(a\) determines how tall or amplified the wave (and consequently the petals) is, and \(m\) influences how frequently the pattern repeats over \(2\pi\).

In \(r = 6 \sin 5 \theta\), the sine function not only shapes the roses but also dictates their orientation and configuration by how \(m\), \(\theta\), and \(r\) interplay, revealing stunning geometrical figures.