Problem 31 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

Graph the curve for \(r = 1 - 2 \sin 3\theta\) using polar coordinates; it forms a 3-petal rose.

Step 1: Understanding the Equation

The equation provided is in the polar form, where \(r = 1 - 2 \sin 3\theta\). This is a type of rose curve with the general form \(r = a \pm b \sin (n\theta)\) or \(r = a \pm b \cos (n\theta)\). This particular equation involves a sinusoidal component with a frequency of 3, which will indicate the number of petals the rose curve will have.

Step 2: Identifying Parameters

For the given equation, \(a = 1\), \(b = 2\), and \(n = 3\). Since \(n\) (the frequency) is odd, the number of petals will be \(3\). The factor \(-2\) in front of the sine function affects the symmetry and orientation of the petals.

Step 3: Setting the Range for \(\theta\)

To graph the entire rose, set the range for \(\theta\) from \(0\) to \(2\pi\). This will ensure that the rose completes one full cycle and all petals are graphed.

Step 4: Plot Points for \(\theta\)

Calculate \(r\) for several values of \(\theta\) within the range \([0, 2\pi]\). Start with \(\theta = 0\), then progress to \(\theta = \pi/6, \pi/3, \ldots, 2\pi\). For each value of \(\theta\), calculate the corresponding \(r\) using the equation \(r = 1 - 2 \sin 3\theta\).

Step 5: Sketch the Polar Graph

Using the calculated points, plot them on a polar coordinate plane. Begin at the pole (origin) and plot each \((r, \theta)\) point. As you move through the sequence of angles from \(0\) to \(2\pi\), draw the curve to connect the points. This will create a pattern resembling a three-petal rose. Note how the distance \(r\) changes, affecting the radial points' length.

Key Concepts

Rose Curve

A rose curve is a type of mathematical curve that looks like a flower with petals. It is represented in polar coordinates by equations of the form \(r = a \pm b \sin(n\theta)\) or \(r = a \pm b \cos(n\theta)\). These curves are fascinating because they create symmetrical patterns depending on the values of \(a\), \(b\), and \(n\).

• In the equation \(r = 1 - 2 \sin 3\theta\), the "3" in \(3\theta\) indicates the function has frequency 3, which affects the number of petals.
• An odd frequency \(n\) results in a rose curve with exactly \(n\) petals.
• Since we have minus in front of 2, the rose curve will flip over the horizontal axis, affecting how these petals extend from the origin.

These rose curves elegantly illustrate how mathematical functions can form beautiful and predictable patterns.

Polar Graph

A polar graph represents equations in the polar coordinate system, where each point on the graph is determined by a radius \(r\) and an angle \(\theta\). This system is different from the Cartesian system, which uses \(x\) and \(y\) coordinates. In polar graphs:

• \(r\) measures the distance from the pole (origin) to a point on the plane.
• \(\theta\) represents the angle from the positive x-axis to the point.
• The polar graph can capture the essence of circular and periodic phenomena like rose curves.

To plot the rose curve for \(r = 1 - 2 \sin 3\theta\), you need to calculate values of \(r\) as \(\theta\) varies from 0 to \(2\pi\). As you sketch the points and connect them, you'll see the inherent symmetry of the curve forming, depicting a flower pattern. This approach highlights how angles and radii work together to visualize complex equations.

Trigonometric Functions

The sine and cosine functions are fundamental to trigonometry and play a pivotal role in creating rose curves. These functions describe periodic, wave-like behavior that oscillates between a set of limits. For example, the equation \(r = 1 - 2 \sin 3\theta\) illustrates this with the sinusoidal behavior of \(\sin 3\theta\).

Components to understand with trigonometric functions in polar equations include:

• The frequency multiplier (in this case, the number 3 in \(3\theta\)) determines the number of oscillation cycles the function completes over the interval from 0 to \(2\pi\).
• The amplitude, driven by the coefficient \(-2\), dictates how far \(r\) from the center of the graph can be, influencing the size of each petal in the rose curve.
• Adjusting these parameters will increase or decrease petal length, affect the curve's range, and change the rotational symmetry.

Trigonometric functions are the heart of dynamic and visually interesting patterns in polar graphs, which are not only beautiful but also mathematically intriguing.