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. $$r=\sin 2 \theta$$

Short answer

To graph the rose with the polar equation \(r=\sin 2\theta\), we first identified important points, determined the range of \(r\) and \(\theta\), and then plotted the rose. The graph is complete with a total of four petals, with two petals occurring between \(0 \leq \theta < \pi\) and two more petals occurring between \(\pi \leq \theta < 2\pi\). The graph will repeat every interval of \(2\pi\).

Step-by-step solution

Identify important points

To create the graph, start by identifying the range of \(r\) and any points where \(r=0\). In this case, \(r=\sin 2\theta\), so it's clear that \(r\) is bounded between \(-1\) and \(1\). By looking at the sine function, we can determine that \(r=0\) when \(2\theta=n\pi\) \((\)where \(n\) is an integer\()\). So, \(\theta=\frac{n\pi}{2}\).

Determine the range of \(\theta\)

Due to the periodic nature of the sine function, we can focus on the range of \(0 \leq \theta \leq \pi\) for our graph. For \(\theta\) in this range, we can observe that: - When \(0 \leq \theta < \frac{\pi}{4}\), \(\sin 2\theta\) is positive and increasing. - When \(\frac{\pi}{4} \leq \theta < \frac{\pi}{2}\), \(\sin 2\theta\) becomes positive but decreasing. - When \(\frac{\pi}{2} \leq \theta < \frac{3\pi}{4}\), \(\sin 2\theta\) is negative but increasing. - And, when \(\frac{3\pi}{4} \leq \theta < \pi\), \(\sin 2\theta\) is negative and decreasing.

Plot the rose

To plot the rose, we can use this analysis and the range of \(r\) values: 1. Begin by plotting the points for \(\theta = 0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4},\) and \(\pi\). It is evident that there are two "petals" between \(0 \leq \theta < \pi\). 2. Repeat this process for the range \(\pi \leq \theta < 2\pi\) since we know that the sine function is periodic with a period of \(2\pi\). This will result in two more petals. 3. Finally, connect the points with a smooth curve to obtain the graph of the rose. 4. Since the period of the sine function is \(2\pi\), the graph will repeat after each interval of \(2\pi\). So, our graph is complete, and we have the rose with polar equation \(r=\sin 2 \theta\).

Key concepts

Rose Curves

A rose curve is a special type of graph often found in polar coordinate systems. These curves resemble the petals of a rose, hence their charming name. A polar equation of the form \(r = a \sin m \theta\) or \(r = a \cos m \theta\), where \(a\) and \(m\) are constants, defines a rose curve. The number of petals produced depends heavily on the value of \(m\). If \(m\) is even, the rose will have \(2m\) petals, whereas if it is odd, it will have \(m\) petals.

Our initial exercise gave us the polar equation \(r = \sin 2 \theta\). Here, \(a = 1\) and \(m = 2\), predicting that the graph will have four petals. These fascinating formations make studying and graphing rose curves a rewarding experience, as they elegantly illustrate the properties of polar equations.

Trigonometric Functions

Delving into trigonometric functions, specifically sine and cosine, is essential when working with rose curves. These functions transform linear angles into the complex, periodic behaviors we observe as curves. The sine function, for instance, oscillates between -1 and 1. This property dictates the maximum and minimum extents of the rose curve.

For the equation \(r = \sin 2 \theta\), the range of \(\sin 2 \theta\) between -1 and 1 directly affects the polar radius \(r\). Since these functions are periodic, the behavior repeats at regular intervals. In our case, this periodic nature translates to an interval of \(2\pi\), which ensures the curve repeats and forms a complete rose. Understanding these sine and cosine characteristics helps in predicting and sketching the behavior of rose curves efficiently.

Graphing Techniques

Graphing polar equations such as rose curves requires some unique techniques compared to Cartesian plotting. Begin by identifying crucial points where the polar radius \(r\) becomes zero, positive, or negative, as these are pivotal to plotting the curve correctly. For \(r = \sin 2 \theta\), \(r\) equals zero at angles where \(2\theta = n\pi\).

Next, divide the interval \(0\) to \(2\pi\) into segments to evaluate \(r\) at specific angles, like \(0, \frac{\pi}{4}, \frac{\pi}{2}, \ldots\). Such points help outline petals, showcasing the number and symmetry properties of the rose.

• Focus on intervals where \(\theta\) makes \(r\) positive, as these will create outward extensions or petals.
• For angles where \(\theta\) leads to negative \(r\), the curve will recede towards or pass through the pole (origin).

Connecting these plotted points with smooth curves generates the entire rose pattern. Once completed for one interval, replications across complete cycles verify the full rose symmetry—often necessitating graph repetition across \(2\pi\) to catch all petals.