Problem 51 - Calculus

Question

(a) Use a graphing calculator or computer to estimate the period of \(2 \sin \theta+3 \cos (2 \theta)\) (b) Explain your answer, given that the period of \(\sin \theta\) is \(2 \pi\) and the period of \(\cos (2 \theta)\) is \(\pi\)

Step-by-Step Solution

Answer

The period of \(2 \sin \theta + 3 \cos (2\theta)\) is \(2\pi\).

Step 1: Identify Function Components

The given function is \(f(\theta) = 2 \sin \theta + 3 \cos (2\theta)\). It is composed of two trigonometric terms: \(2 \sin \theta\) and \(3 \cos (2\theta)\). We need to analyze each part separately to understand their contributions to the function's period.

Step 2: Determine Individual Periods

The period of \(\sin \theta\) is \(2\pi\), meaning that it completes one full cycle as \(\theta\) goes from 0 to \(2\pi\). The period of \(\cos (2\theta)\) is \(\pi\) because the factor of 2 inside the cosine function compresses the period to \(\frac{2\pi}{2} = \pi\).

Step 3: Find the Least Common Multiple of Periods

To find the period of the entire function \(f(\theta) = 2 \sin \theta + 3 \cos (2\theta)\), we must determine when both components complete a full cycle simultaneously. This happens at the least common multiple (LCM) of the individual periods: \(2\pi\) and \(\pi\).

Step 4: Calculate the Least Common Multiple

The LCM of \(2\pi\) and \(\pi\) is \(2\pi\), since \(2\pi\) is evenly divisible by \(\pi\). Thus, both components complete their cycles together every \(2\pi\).

Step 5: Verify with a Graphing Tool

Use a graphing calculator or software to plot \(f(\theta) = 2 \sin \theta + 3 \cos (2\theta)\). Verify that the pattern of the graph repeats every \(2\pi\), confirming that the period of the function is \(2\pi\).

Key Concepts

Periodicity

When we talk about **periodicity** in trigonometric functions, we mean the interval after which the function repeats its values. This concept is vital for understanding and analyzing periodic waves like those described in functions such as sine and cosine. These functions, by their nature, have a set length in their cycle that is consistent and predictable.

The period of a function is the smallest interval over which the values repeat. For a simple sine function, \( \sin \theta \), this period is \( 2\pi \), meaning every time \( \theta \) increases by \( 2\pi \), the function starts its cycle again.Exploring complex functions like \( f(\theta) = 2 \sin \theta + 3 \cos(2\theta) \), it's essential to find when both individual components complete their cycles. This is determined by calculating the least common multiple (LCM) of their periods, which reveals the period of the overall function.

• The LCM ensures that the complex function’s cycles align and repeat predictably.
• For any trigonometric function combination, understanding periodicity helps simplify analysis and predictions.

Sine Function

The **sine function** is a foundational part of trigonometry. It's known for its characteristic wave-like pattern, stretching infinitely in both directions. Its graph is a smooth, continuous curve that oscillates between -1 and 1.

The basic sine function, \( \sin \theta \), has a period of \( 2\pi \). This means it takes a full \( 2\pi \) radians for the sine function to complete a cycle and start repeating its values. Each cycle has:

• Two intercepts with the horizontal axis.
• One maximum value at 1.
• One minimum value at -1.

When you combine the sine function with other functions, like in \( f(\theta) = 2 \sin \theta + 3 \cos(2\theta) \), the amplitude may change (due to multiplication by a coefficient like 2) but its period will still affect the combined function significantly.

The sine function's contribution to periodicity in complex functions is crucial and is a primary component in understanding more intricate trigonometric phenomena.

Cosine Function

Similar to its cousin the sine function, the **cosine function** also features prominently in trigonometry with its own unique wave-like behavior. The core cosine function, \( \cos \theta \), mirrors the sine curve but is shifted by \( \frac{\pi}{2} \) radians to the left.

Its period, like that of the sine function, is \( 2\pi \). This means that, over an interval of \( 2\pi \), the values will repeat in a regular pattern. However, when we see \( \cos(2\theta) \), the factor of 2 in front of \( \theta \) compresses the cycle, creating a period of \( \pi \). This compression means the values repeat twice as fast compared to the usual cosine function.

• For \( \cos(2\theta) \), the graph completes its cycle from peak to peak twice as quickly.
• Understanding the compressed period helps in pinpointing regular repetitions in more complex functions.

In combination functions like \( f(\theta) = 2 \sin \theta + 3 \cos(2\theta) \), the cosine component modifies the periodicity, cueing us to calculate the LCM of individual components’ periods to find the overall function's period.

Exploring how these modifications affect the broader function's behavior helps in mastering trigonometric identities and their applications.