Question

Equations of the form \(r=a \sin m \theta\) or \(r=a \cos m \theta,\) where \(a\) is a real number and \(m\) is a positive integer, have graphs known as roses (see Example 6 ). Graph the following roses. \(r=4 \cos 3 \theta\)

Short answer

Answer: The rose plot has 3 petals, and they are symmetric with respect to the \(\theta = 0\) line (or x-axis).

Step-by-step solution

Find the number of petals

Let's analyze the given polar equation: \(r = 4\cos{3\theta}\). According to the general form, \(r = a\cos{m\theta}\) or \(r = a\sin{m\theta}\), we can identify the parameters of our equation: - \(a=4\), the length of the petals - \(m=3\), the number of petals when the equation is in cosine form Since our equation is in cosine form, we have \(3\) petals.

Calculate Roses Orientation

A rose plot equation with cosine form has petals symmetric with respect to the \(\theta = 0\) line (or x-axis). Since our equation is in cosine form, the rose plot will have a petal on \(\theta = 0\) line.

Create a table of values

It's helpful to create a table of values for \(\theta\) and \(r\). We can choose a step size for \(\theta\), such as \(\frac{\pi}{12}\), and determine the corresponding \(r\) values. Here's a sample of such a table: | \(\theta\) | \(r\) | | -------- | --- | | \(0\) | \(4\) | | \(\frac{\pi}{12}\) | \(2\) | | \(\frac{\pi}{6}\) | \(0\) | | \(\frac{5\pi}{12}\) | \(2\) | | \(\frac{\pi}{2}\) | \(4\) | | \(\frac{7\pi}{12}\) | \(2\) | This should be perfect to start graphing our polar equation.

Graph the Polar Equation

Using the table of values we created in the previous step, graph the polar equation on a polar grid. Start by plotting the points on the grid and then connecting those points to draw the petals of the rose. Remember that the petals are symmetric with respect to the \(\theta=0\) line, so you will have a petal on that line. You will get a total of 3 petals as determined in Step 1, which confirms our initial analysis. Review the graph to ensure that the shape and orientation satisfy the properties of the given polar equation \(r=4\cos{3\theta}\).

Key concepts

Roses

In polar coordinates, roses are a fascinating type of graph that emerge from specific trigonometric equations. The formula for these graphs is generally given by \(r = a \sin m \theta\) or \(r = a \cos m \theta\). Here, \(r\) is the radial distance, \(a\) is the amplitude (dictating the length of the petals), and \(m\) is a positive integer that influences the number of petals. These beautiful symmetric patterns often resemble actual flower petals, hence the name "roses."

The number of petals in a rose curve depends on whether the equation uses sine or cosine and whether \(m\) is even or odd. When \(m\) is odd, the number of petals is \(m\). For even \(m\), the sine and cosine equations produce \(2m\) petals. For our example, \(r = 4 \cos 3 \theta\), it's a cosine equation with \(m=3\), leading to three petals.

Roses are not only aesthetically appealing but also serve as an excellent way to explore the concepts of polar equations and symmetry in mathematics.

Petals

The petals of a rose in polar coordinates represent key components of the graph's beautiful structure. Each petal's length is determined by the coefficient \(a\) in the equations \(r = a \sin m \theta\) or \(r = a \cos m \theta\). Therefore, with an equation like \(r = 4 \cos 3 \theta\), each petal reaches its maximum length of \(4\), the absolute value of \(a\).

For cosine-based equations, like \(r = 4 \cos 3 \theta\), the petals are arranged symmetrically around the horizontal axis, meaning one petal aligned along \(\theta = 0\). In contrast, sine-based equations align petals along the vertical axis. The angle \(\theta\) at which each petal is centered can be found by distributing the total angular coverage (based on \(m\)) around a complete circle.

Petals are often equal in size and symmetrically placed, adding to the charm of rose curve graphs.

Trigonometric Functions

Trigonometric functions like sine and cosine are fundamental to understanding polar coordinates and the graphs they produce. These functions help map angles to corresponding radial distances, either through \(r = a \sin m \theta\) or \(r = a \cos m \theta\), creating various shapes, including roses.

In the context of polar equations, the trigonometric function describes how the radius \(r\) changes as the angle \(\theta\) changes. Cosine functions, as in \(r = 4 \cos 3 \theta\), start at a maximum when \(\theta = 0\) and model symmetric graphs around the horizontal axis. Sine functions, however, introduce a phase shift, positioning graphs symmetrically around the vertical axis.

Understanding these functions' properties, such as periodicity and amplitude, is essential. It helps predict the number of petals and their alignment within a rose, contributing to accurate graph creation.

Graphing Polar Equations

Graphing polar equations, like those of roses, involves unique techniques compared to Cartesian graphs. Using the polar coordinate system, each point is determined by an angle \(\theta\) and a distance \(r\) from the origin.

To graph an equation like \(r = 4 \cos 3 \theta\), you start by determining key points using a table of values for \(\theta\). Choose strategic angles that demonstrate the pattern, such as \(\theta = 0\), \(\frac{\pi}{6}\), or \(\frac{\pi}{2}\). For each \(\theta\), calculate \(r\) to locate points on the polar grid.

• Plot each point by rotating \(\theta\) degrees from the polar axis and moving outwards or inwards by \(r\).
• Connect the plotted points smoothly to outline the petals.
• Consider the symmetry inherent in the trigonometric form; if \(r = a \cos m \theta\), expect symmetry around the horizontal axis.

Graphing polar equations highlights the relationship between angles and distances, resulting in elegant geometric patterns like rose curves. Mastery of this process enriches understanding of polar coordinates and the beauty they can illustrate.