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

Question: Graph the polar equation $$r = 4\cos(3\theta)$$ and describe its characteristics. Answer: The graph of the polar equation $$r = 4\cos(3\theta)$$ is a rose curve with 3 petals. This is because the value of m in the equation is 3, which is an odd number. The rose curve has 3 petals, and each petal is connected smoothly to form the complete graph.

Step-by-step solution

Identify the characteristics of the given polar equation

We have the equation $$r = 4 \cos(3\theta)$$. In this equation, we can see that a = 4 and m = 3. The rose graph has properties dependent on the value of m: - If m is odd, there will be m petals in the graph. - If m is even, there will be 2m petals in the graph. Since m = 3, which is odd, our graph will have 3 petals.

Set up a table to find the values of r for specific values of θ

Create a table to find the values of r for different values of θ. We can choose values between 0 and 2π (as the graph repeats after 2π). | θ(rad) | r | |:------:|:--| |0 | 4 | |π/3 | 0 | |2π/3 | -4| |π | 0 | |4π/3 | 4 | |5π/3 | 0 | |2π | 4 |

Plot the polar coordinates on a polar coordinate plane

Now we need to plot the following coordinates on a polar coordinate plane: 1. (4, 0) 2. (0, π/3) 3. (-4, 2π/3) 4. (0, π) 5. (4, 4π/3) 6. (0, 5π/3) 7. (4, 2π) Remember that our angle θ should be measured in radians.

Draw the graph of the rose curve

Once the points have been plotted on the polar coordinate plane, we connect the points smoothly to form the rose curve. In this case, we'll have 3 petals in the graph. And there you have it: the graph of the rose curve for the polar equation $$r = 4\cos(3\theta)$$, which has 3 petals.

Key concepts

Rose Curves

Rose curves are a fascinating family of graphs encountered in polar coordinates. They are characterized by their petal-like shapes and can be defined by the equations \( r = a \sin m \theta \) or \( r = a \cos m \theta \), where \( a \) is a constant and \( m \) is a positive integer. The numbers of petals in a rose curve hang on the value of \( m \). If \( m \) is an odd number, the curve will have \( m \) petals. On the other hand, if \( m \) is even, the curve will boast \( 2m \) petals. The variable \( a \) influences the size of each petal.

To visualize rose curves, it's crucial to understand the role played by the trigonometric function used in the formula. The formula with \( \sin \) will generate petals oriented differently compared to those from \( \cos \), yet they share the basic property of having symmetrical petals distributed evenly around the origin.

These curves are popular in mathematics due to their striking, floral symmetry and are used educationally to explore concepts in polar coordinates and trigonometry.

Trigonometric Functions

Trigonometric functions are foundational to understanding both polar coordinates and rose curves. These functions, including sine and cosine, describe the relationships between the angles and sides of right triangles. They maintain periodic properties, which means they repeat their values in regular intervals, such as every \( 2\pi \) radians for sine and cosine.

In polar equations like \( r = a \cos m \theta \) and \( r = a \sin m \theta \), the trigonometric functions determine the circular nature of the resulting graph. The value \( \theta \) represents the angle, while the function's amplitude affects the radius at different points. This is why, in rose curves, you see petals extending from the origin following the wave nature of these functions.

Understanding these functions helps to predict the shape and behavior of polar graphs, an essential skill in fields such as physics and engineering.

Graphing Polar Equations

Graphing polar equations is a unique endeavor due to their use of polar coordinates. Unlike Cartesian graphs, polar graphs are guided by an angle \( \theta \) and a radius \( r \). In polar coordinate systems, points are plotted using these two parameters, which often create circular and spiral patterns.

When graphing an equation like \( r = 4 \cos 3 \theta \), it is important to:

• Set a range for \( \theta \), typically from \( 0 \) to \( 2\pi \), given the periodic nature of trigonometric functions used in polar equations.
• Calculate \( r \) for various values of \( \theta \) within the range to determine the coordinates to plot.
• Use a polar graph, plotting each set of \( (r, \theta) \) coordinates with respect to the origin.
• Connect these points smoothly to view the resulting curve, keeping an eye on patterns, such as the number of petals in a rose curve.

Graphing polar equations requires practice and understanding of underlying principles like symmetry and periodicity, which are essential to mastering this distinctive mathematical visualization.