Problem 79 - Precalculus

Question

Consider the equation \(r=3 \sin k \theta\). (a) Use a graphing utility to graph the equation for \(k=1.5 .\) Find the interval for \(\theta\) over which the graph is traced only once. (b) Use the graphing utility to graph the equation for \(k=2.5 .\) Find the interval for \(\theta\) over which the graph is traced only once. (c) Is it possible to find an interval for \(\theta\) over which the graph is traced only once for any rational number \(k ?\) Explain.

Step-by-Step Solution

Answer

The interval for \(\theta\) which traces the graph only once when \(k=1.5\) is \(0 \leq \theta < 4\pi/3\), and when \(k=2.5\) is \(0 \leq \theta < 4\pi/5\). It is possible to find an interval for \(\theta\) over which the graph is traced only once for any rational \(k\), using the interval \(0 \leq \theta < 2\pi/k\).

Step 1: Graph for k = 1.5

Using a graphing tool, plot the given equation \(r=3sin(1.5\theta)\). Observe that the graph forms a rose with 3 leaves. The graph of a rose is traced only once for the interval \(0 \leq \theta < 2\pi / k\). Hence, for \(k=1.5\), this interval is \(0 \leq \theta < 4\pi/3\).

Step 2: Graph for k = 2.5

Next, plot the equation \(r=3sin(2.5\theta)\). The graph exhibits 5 leaves this time. Using the same interval and substitution rule, the interval for \(\theta\) for which the graph is traced only once is \(0 \leq \theta < 2\pi / 2.5 = 0 \leq \theta < 4\pi / 5\).

Step 3: Discussion of rational k

Finally, the question states if it's possible to find an interval over which the graph is traced only once for any rational number \(k\). The answer is yes. Irrespective of the value of \(k\), it can be a whole or a fractional number, the interval \(0 \leq \theta < 2\pi / k\) ensures that the graph is traced only once. If \(k\) is rational, the graph will simply repeat the pattern of petals multiple times within the interval \(0 \leq \theta < 2\pi\).

Key Concepts

Rose Curves

Rose curves are a fascinating type of polar curve characterized by a floral pattern of petals or leaves radiating from the origin. They are depicted using the equation \( r = a \sin(k\theta) \) or \( r = a \cos(k\theta) \), where \( a \) and \( k \) are constants. The parameter \( k \) determines the number of petals in the curve.

• If \( k \) is a whole number, and \( k \) is even, the curve will have \( 2k \) petals.
• If \( k \) is a whole number, and \( k \) is odd, the curve will have \( k \) petals.
• If \( k \) is a fractional or rational number, the definition of the petals' number is more complex, often showing a partially overlapping cycle due to non-integer repetitions.

The shape and symmetry of rose curves make them a captivating subject of study in polar coordinates. They provide visually appealing demonstrations of mathematical concepts and can help deepen the understanding of trigonometric functions when visualized.

Graphing Utility

Graphing utilities are powerful tools that help visualize equations, particularly useful for understanding complex curves like rose curves. They can be found as standalone software, online platforms, or calculator features. By plotting equations like \( r=3\sin(k\theta) \), users can explore how changes in parameters affect the graph.
To create a polar plot using a graphing utility:

• Input the equation in polar form, ensuring to set the mode to interpret angles in radians.
• Adjust the viewing window to ensure the entire graph is visible.
• Observe the structure and formation of the graph as different values of \( k \) are used. This visual aid can significantly enhance comprehension of how the curve behaves.

Graphing utilities are indispensable for students learning how various polar curves are traced by mathematical equations over specified angle intervals. This visual representation helps in connecting theoretical concepts to practical visual outputs.

Rational Numbers

Rational numbers, those that can be expressed as the quotient or fraction \( \frac{m}{n} \) of two integers \( m \) and \( n \), with \( n eq 0 \), play a crucial role in the formation of rose curves. When \( k \) is a rational number in the equation \( r=3\sin(k\theta) \), it can represent fractional petal formations in rose curves.
The choice of \( k \) dictates the periodic nature of the curve:

• For rational \( k \), the rose curve will repeat itself in intervals of \( 2\pi/k \), tracing the entire graph once over that range.
• This classification aids in understanding the symmetry and coverage of biological or natural patterns similar to rose curves.

Understanding the influence of rational numbers on polar equations allows learners to appreciate their continuity and how they create repetitive periodic patterns from seemingly simple expressions.

Trigonometric Equations

Trigonometric equations are equations involving trigonometric functions like sine, cosine, and tangent. In the context of rose curves, trigonometric equations of the form \( r = 3\sin(k\theta) \) provide a foundation for understanding how angle and radius relate in polar coordinates.
Key components of these trigonometric equations include:

• Amplitude \( a \): Determines the length of the petals in a rose curve—the bigger \( a \), the longer the petals.
• Frequency \( k \): Dictates the number of petals. Higher \( k \) values result in a denser appearance as the frequency of cycles around the axis increases.
• Phase shifts can also be considered by adding a constant (e.g., \( r = 3\sin(k\theta + c) \)), which would rotate the curve along the \( \theta \) angle.

By solving these equations, students gain insight into periodicity and symmetry inherent in trigonometric functions. This knowledge is critical when exploring and crafting more complex graphical representations of mathematical relationships.