Question

Use a graphing utility to graph the Following equations. In each case, give the smallest interval \([0, P]\) that experates the entire curve (if possible). $$r=\cos 3 \theta+\cos ^{2} 2 \theta$$

Short answer

Answer: The smallest interval [0, P] is [0, 6π].

Step-by-step solution

Identify the trigonometric functions

The given polar equation is \(r = \cos 3\theta + \cos^2 2\theta\). We have two trigonometric functions: the cosine function with an angle of \(3\theta\) and \(2\theta\).

Determine the periods of trigonometric functions

The period of \(cos(k\theta)\), where \(k>0\), is \(\frac{2\pi}{k}\). In first function, we have \(k=3\), and for the second function, we have \(k=2\). So, the periods of these functions are \(\frac{2\pi}{3}\) and \(\pi\).

Find the least common multiple of the periods to find the period of the entire polar equation

By determining the least common multiple (LCM) of \(\frac{2\pi}{3}\) and \(\pi\), we can find the smallest interval that captures the entire polar curve. The periods are 2 and 3 multiples of \(\pi\), so the LCM is 6: \(LCM\left(\frac{2\pi}{3},\pi\right)=6\pi.\)

Use a graphing utility to graph the polar equation

At this point, you need to use a graphing utility like Desmos, GeoGebra, or a graphing calculator to plot the polar equation, \(r = \cos 3\theta + \cos^2 2\theta\), for the interval \(0 \leq \theta \leq 6\pi\). This interval repeats the entire polar curve without missing any part of it.

Determine the interval needed to generate the entire curve

After graphing the polar equation, we can confirm that to generate the entire polar curve, we can use the interval \([0, 6\pi]\). In conclusion, the smallest interval \([0, P]\) that generates the entire curve of the given polar equation, \(r = \cos 3\theta + \cos^2 2\theta\), is \([0, 6\pi]\).

Key concepts

Trigonometric Functions

Trigonometric functions play a crucial role in calculus and geometry, especially when graphed in polar coordinates. In the equation \(r = \cos 3\theta + \cos^2 2\theta\), we encounter two such functions: \(\cos 3\theta\) and \(\cos 2\theta\).

Cosine functions express the horizontal component of a point on the unit circle, varying from -1 to 1. It repeats this pattern as \(\theta\) increases. Understanding these trigonometric functions helps us predict curve behavior, allowing us to explore complex graphs.

• \(\cos(k\theta)\): Changes frequency based on \(k\) value.
• \(\cos^2(\theta)\): Modifies amplitude and period compared to \(\cos \theta\).

Recognizing these features lets us tackle polar equations more efficiently, which is invaluable when solving similar problems.

Periodicity

Periodicity refers to how trigonometric functions repeat their values over regular intervals. For \(\cos(k\theta)\), the period is \(\frac{2\pi}{k}\), indicating the length before it starts repeating.

In our equation, the periodic nature shows in \(\cos 3\theta\) with a period of \(\frac{2\pi}{3}\), and \(\cos^2 2\theta\) aligns with \(\pi\). These periods give us insight into repetition points.

Understanding periodicity allows us to:

• Predict graph cycles and shapes
• Find common intervals that encompass full cycles of the equation

The combined periodic patterns offer an effective way to explore polar curves' properties, ensuring we capture every aspect of the graph.

Graphing Utility

Graphing utilities are powerful tools that visually represent equations, making complex concepts tangible. Using software like Desmos or GeoGebra, we can graph functions like the polar equation \(r = \cos 3\theta + \cos^2 2\theta\).

By inputting the equation, visualize its curve in the polar coordinate system, simplifying the analysis of its periodic features. Here's how it can help:

• Plotting accurately for specific intervals
• Zooming in for detailed observation
• Comparing multiple functions for study

The ability to explore a complete equation graphically is essential for understanding the full behavior of trigonometric functions and developing the intuition necessary for topics like periodicity.

Least Common Multiple

When working with multiple periodic functions, finding the least common multiple (LCM) of their periods allows us to determine an interval capturing the entire curve. For the polar equation \(r = \cos 3\theta + \cos^2 2\theta\), the periods \(\frac{2\pi}{3}\) and \(\pi\) have an LCM of \(6\pi\).

This LCM helps us find the smallest interval \([0, 6\pi]\) where the graph completes one full pattern, representing all features exhaustively.

Understanding LCM's role in:

• Aligning different periodic functions
• Ensuring full feature representation
• Reducing repetitive calculations

Enables efficient graph analysis and problem-solving, especially in the realm of polar equations.