Problem 69 - Calculus

Question

Are the statements true or false? Give an explanation for your answer. The function \(g(\theta)=e^{\sin \theta}\) is periodic.

Step-by-Step Solution

Answer

True, the function is periodic with period \(2\pi\).

Step 1: Identify the Definition of a Periodic Function

A function is periodic if there exists a non-zero constant \( T \), such that \( g(\theta + T) = g(\theta) \) for all values of \( \theta \).

Step 2: Examine \( g(\theta) = e^{\sin \theta} \) for Periodicity

The function \( g(\theta) = e^{\sin \theta} \) involves the exponential function and the sine function. The sine function \( \sin \theta \) is periodic with period \( 2\pi \). Therefore, \( \sin(\theta + 2\pi) = \sin(\theta) \).

Step 3: Apply the Periodicity of the Sine Function

Since \( \sin(\theta + 2\pi) = \sin(\theta) \), it follows that \( e^{\sin(\theta + 2\pi)} = e^{\sin \theta} \). This implies that \( g(\theta+2\pi) = e^{\sin(\theta + 2\pi)} = e^{\sin \theta} = g(\theta) \).

Step 4: Confirm the Period of the Function

Calculating the function \( g(\theta) \) at intervals of \( \theta + 2\pi \) confirms the above steps and shows that the function repeats its values every \( 2\pi \). Hence, the function is periodic with period \( 2\pi \).

Key Concepts

Understanding the Sine Function

The sine function, denoted as \( \sin \theta \), is one of the fundamental trigonometric functions that describe the relationship between angles and the ratios of sides in a right triangle. One of the key properties of the sine function is its periodicity. This means that it repeats its values at regular intervals. For the sine function, this interval is \( 2\pi \). This implies that for any angle \( \theta \), adding \( 2\pi \) results in the same value of the sine function.

• Periodic interval: \( 2\pi \)
• Pattern repeats: \( \sin(\theta + 2\pi) = \sin(\theta) \)

Because of its periodic nature, the sine function forms the basis for understanding more complex periodic behaviors seen in various mathematical functions and real-world phenomena. Being periodic with period \( 2\pi \) means that every complete rotation or cycle returns the function to its starting point, producing a smooth and continuous wave-like pattern.

What is an Exponential Function?

Exponential functions involve a constant base raised to a variable exponent. A simple example is \( e^x \), where \( e \) is the base of the natural logarithm, approximately equal to 2.71828. These functions grow rapidly and are not limited or restricted by periodicity in the same way that trigonometric functions are. Unlike sine functions, exponential functions do not have a regular repeating pattern and can be continuous and unbounded.

• Rapid growth: Exponential functions increase at an accelerating rate.
• Non-periodic: They do not repeat their values in a regular cycle.

The exponential function \( e^{\sin \theta} \) combines both a non-periodic exponential and a periodic sine component. This combination creates a new function with interesting properties, including periodicity from its trigonometric part, while maintaining the exponential growth characteristic of \( e^x \).

Exploring Function Periodicity

Periodicity in functions is an important concept in mathematics that involves a function repeating its values at fixed intervals. A function \( f(x) \) is periodic if there exists a non-zero constant \( T \) such that for every input \( x \), the output of the function remains the same: \( f(x + T) = f(x) \).

• If \( T \) is the smallest positive constant for which this is true, \( T \) is the period of the function.
• Periodic functions include waves and cycles seen both in nature and in mathematical models.

For the function \( g(\theta) = e^{\sin \theta} \), the periodic \( \sin \theta \) function influences the overall periodicity, resulting in a period of \( 2\pi \). Hence, despite the involved exponential component \( e^{\sin \theta} \) itself not being periodic, the overall function inherits periodic behavior due to the sine component. This unique combination results in a function that repeats every \( 2\pi \), confirming it's periodic with that period.