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. $$r=4 \cos 3 \theta$$

Short answer

Question: Sketch the graph of the polar equation $$r=4\cos 3\theta$$. Answer: 1. Identify the range of θ: The range to consider is \(0 \le \theta \le \pi\). 2. Determine the key points: Intersection points with θ-axis are \(\theta_1=\frac{\pi}{6}\) and \(\theta_2=\frac{\pi}{2}\). The petal tip is at \(\theta_t = 0\). 3. Plot the key points in polar coordinates: Plot (\(\theta_0=0\), \(r_0=4\)), (\(\theta_1=\frac{\pi}{6}\), \(r_1=0\)), and (\(\theta_2=\frac{\pi}{2}\), \(r_2=0\)). 4. Trace the graph smoothly, with 3 petals and symmetry around the y-axis.

Step-by-step solution

Identify the range of θ

For graphing in polar coordinates, we usually look at the range \(0 \le \theta \le 2\pi\). However, when dealing with rose curves, we need to take into account the number of petals (m). In this case, m=3. Since we have an even function (cosine), the petals will be symmetrical around the y-axis, hence we only need to consider the range of \(0 \le \theta \le \pi\) instead of \(0 \le \theta \le 2\pi\).

Find the key points

We'll find the key points when the polar curve intersects the θ-axis and at the tip of each petal. This will occur when r=0 and when r is maximum (4 in our case). For r=0, $$0=4\cos 3\theta$$ $$\cos 3\theta = 0$$ Now we can find the values of θ in our range that satisfy this equation: $$3\theta = \frac{\pi}{2}, \frac{3\pi}{2}$$ So, we have two intersection points: $$\theta_1 = \frac{\pi}{6}, \;\; \theta_2 = \frac{\pi}{2}$$ For r=4 (maximum), $$4=4\cos 3\theta$$ $$\cos 3 \theta = 1$$ Now we can find the values of θ in our range that satisfy this equation: $$3\theta = 0, 2\pi$$ So, we have one tip point: $$\theta_t = 0$$ Therefore, the key points are: $$\theta_0 = 0, \;\; \theta_1 = \frac{\pi}{6}, \;\; \theta_2 = \frac{\pi}{2}$$ with their corresponding r values: $$r_0 = 4, \;\; r_1 = 0, \;\; r_2 = 0$$

Graph the points in polar coordinates

Now that we have the key points, we can start graphing our polar curve. Begin by marking \(\theta_0 = 0\) and the corresponding r value \(r_0 = 4\). Continue with \(\theta_1 = \frac{\pi}{6}\) and the corresponding r value \(r_1 = 0\). Finally, plot \(\theta_2 = \frac{\pi}{2}\) and the corresponding r value \(r_2=0\).

Trace the graph smoothly

With the key points plotted, we can now trace the graph smoothly. Remember that the graph will have a symmetry around the y-axis and 3 petals. Trace the graph such that it follows the key points and maintains the symmetrical properties for this rose curve. Now you should have the graph of the following rose curve polar equation: $$r=4\cos 3\theta$$

Key concepts

Rose Curves

Rose curves are a fascinating example of polar graphs that present a floral-like shape with distinct petals. These curves are generated through polar equations of the form \(r = a \sin m \theta\) or \(r = a \cos m \theta\). Here, \(a\) determines the length of the petals, and \(m\) - a positive integer - affects the number of petals displayed. Indeed, if \(m\) is odd, the rose will have \(m\) petals, while an even \(m\) will yield \(2m\) petals.
The distinctive beauty of rose curves lies in their symmetry and repetitive pattern, often used as an engaging way to merge algebra with geometry. As you graph, note the symmetrical properties which mirror around the polar axes. When graphed, each petal emerges as \(\theta\) progresses, and the curve reflects itself at critical points.
Understanding rose curves introduces students to comprehending periodic behavior and symmetry in larger mathematical contexts, paving the way for further studies in trigonometry and calculus.

Graphing Polar Equations

Graphing polar equations can seem complex at first, but simplifying the process step-by-step makes this task much more approachable. Unlike Cartesian coordinates, which divide the plane into a grid, polar coordinates focus on the angle \(\theta\) and the distance \(r\) from the origin.
To start graphing, identify the function type and determine its symmetry. For the exercise equation \(r=4\cos 3\theta\), the cosine functions offer symmetry about the polar axis. Next, find the critical points: where \(r=0\) and where \(r\) achieves its maximum value. These points often guide how the curve will behave.
In our example, graphing is smoothly executed by plotting these crucial points first and then connecting them considering symmetry. The unique characteristic of polar graphs is that they can revolve multiple times around the origin, offering visual appeal and practical richness in real-world applications.

Trigonometric Functions

Trigonometric functions are the backbone of rose curves and play a significant role in the world of polar coordinates, weaving between algebra and geometry. Within this context, functions like \(\sin\) and \(\cos\) define not only the circular rotations but also oscillations that create wave-like appearances.

• Cosine Function: In rose curves, the cosine function determines the starting orientation of the first petal and reflects all other petals symmetrically.
• Sine Function: Similarly, a sine function would produce rose curves with petals offset from the polar axis, adding a different symmetrical dimension.

The intricate dance of these functions allows for beautifully complex graphs, enabling learners to visualize concepts that are just formulas on paper. Understanding trigonometric functions in polar equations enhances one's ability to tackle problems in physical sciences, sound waves, and rhythmic phenomena, where periodicity and symmetry go hand in hand. This foundation is crucial for more advanced mathematical explorations and applications.