Problem 45 - Thomas Calculus

Question

A rose within a rose Graph the equation \(r=1-2 \sin 3 \theta\)

Step-by-Step Solution

Answer

The polar equation \(r = 1 - 2 \sin 3\theta\) is a rose curve with 3 petals, symmetric about the origin.

Step 1: Identify the Type of Polar Equation

The given equation is in the form of a polar equation: \(r = 1 - 2 \sin 3\theta\). This is a rose curve, commonly characterized by the form \(r = a + b \sin(n\theta)\) or \(r = a + b \cos(n\theta)\). In this case, it's a rose curve with three petals (since the coefficient of \(\theta\) is 3).

Step 2: Determine Key Features of the Curve

For the equation \(r = 1 - 2 \sin 3\theta\), we can determine the number of petals and symmetry. Since \(n=3\) is odd and \(b=2\) which is greater than \(a=1\), the number of petals will be \(3\) for \(\sin(n\theta)\). Additionally, the curve will have rotational symmetry about the origin.

Step 3: Calculate Critical Points

To graph the polar curve, identify critical points where the curve intersects the origin by setting \(r=0\). Solving \(1 - 2 \sin 3\theta = 0\) gives \(\sin 3\theta = \frac{1}{2}\). The general solutions for \(3\theta\) where \(\sin(3\theta) = \frac{1}{2}\) are \(3\theta = \frac{\pi}{6} + 2k\pi, \frac{5\pi}{6} + 2k\pi\). Solve for \(\theta\) to find values that help in plotting the graph.

Step 4: Plotting the Graph

Plot the polar equation by using the critical points and features calculated above. With symmetry about the origin, the petals will be evenly spaced. Start with the angle \(\theta = \frac{\pi}{18}, \frac{\pi}{2}, \frac{5\pi}{18}\), rotate them appropriately at equal intervals until \(\theta = 2\pi\). This generates the looping rose petal shape.

Step 5: Drawing and Verifying

Verify the smoothness and periodicity of the graph by checking other points. Evaluate \(r\) at multiple \(\theta\) values to ensure the graph is accurate and the petals are correctly shaped and positioned. Check for any repeating pattern, highlighting its rotational nature.

Key Concepts

Rose Curve

A rose curve is a special type of polar graph that is celebrated for its captivating petal-like patterns. These curves arise from polar equations of the form \( r = a + b \sin(n\theta) \) or \( r = a + b \cos(n\theta) \), where \( n \) dictates the number of petals. If \( n \) is odd, the curve will have \( n \) petals, whereas if \( n \) is even, it will display \( 2n \) petals.
In the equation \( r = 1 - 2 \sin 3\theta \), you notice \( n = 3 \), resulting in a three-petaled rose curve due to the odd coefficient. This equation also displays variations in amplitude with \( a = 1 \) and \( b = -2 \), which influences the curve’s range throughout its rotation, guiding the overall size and spacing of the petals.
Rose curves offer an infinite array of visual appeal and are commonly seen in mathematical constructs, coded designs, and occasionally even in nature.

Graph of Polar Equations

Graphing polar equations like rose curves introduces an exciting challenge. Polar coordinates use the radius \( r \) and angle \( \theta \) to determine points on a plane, which is a different approach from the rectangular coordinate system.
To graph \( r = 1 - 2 \sin 3\theta \), focus on understanding how \( r \) changes as \( \theta \) varies. The equation guides you on calculating critical points where \( r = 0 \), setting the curve to intersect the origin. Solving these provides fundamental points for plotting your graph.

• Set \( r = 0 \): Solve for \( \theta \) to establish key intersection points.
• Analyze \( \theta \) at intervals: Check varying \( \theta \) values such as \( \frac{\pi}{18}, \frac{\pi}{2}, \frac{5\pi}{18} \), and so forth, to map out the rose.
• Keep symmetry in mind: The beauty of these curves lies in their periodicity.

This method produces the looping and petal-shaped pattern associated with rose curves, superbly illustrating the graph of the polar equation.

Symmetry in Polar Graphs

Symmetry is a fundamental concept when graphing polar equations, offering insights into how these curves repeat and overlap. Polar graphs often demonstrate symmetry about one or more axes or even about the origin; understanding these symmetries can aid both in analyzing equations and graphing them efficiently.
For our rose curve \( r = 1 - 2 \sin 3\theta \), rotational symmetry about the origin is evident. This symmetry means the graph looks the same when rotated by certain angles – this is consistent with the petal-like quality of rose curves. Each petal is identical in shape and size.
To leverage symmetry in graphing:

• Identify possible symmetrical properties of the function.
• Use symmetry to reduce the computational workload by only calculating necessary portions of the graph.
• Ensure accurate plotting by checking repetitive patterns across the full \( 2\pi \) interval.

Recognizing and utilizing symmetry not only speeds up the graphing process but also enriches your understanding of the intrinsic beauty of polar graphs.