Problem 61 - Calculus

Question

Are the statements true or false? Give an explanation for your answer. The function \(f(t)=\sin (0.05 \pi t)\) has period 0.05

Step-by-Step Solution

Answer

The statement is false; the period is 40, not 0.05.

Step 1: Understanding periodic functions

A function is periodic if there exists a non-zero constant \(T\) such that \(f(t + T) = f(t)\) for all values of \(t\). The smallest positive \(T\) that satisfies this condition is called the period of the function.

Step 2: Identifying the period of the sine function

The sine function \(\sin(x)\) has a fundamental period of \(2\pi\). This means \(\sin(x) = \sin(x + 2\pi)\).

Step 3: Determining the period of \(f(t) = \sin(0.05\pi t)\)

To find the period \(T\) for \(f(t) = \sin(0.05\pi t)\), set up the equation based on the sine function's period, so \(0.05\pi(t + T) = 0.05\pi t + 2\pi\). Simplifying gives \(0.05\pi T = 2\pi\).

Step 4: Solving for the period \(T\)

From \(0.05\pi T = 2\pi\), we can divide both sides by \(0.05\pi\): \[T = \frac{2\pi}{0.05\pi} = \frac{2}{0.05} = 40\]. So the period of \(f(t)\) is 40.

Step 5: Conclusion on the given statement

The statement claims that the period is 0.05, but our calculation shows that the actual period is 40. Therefore, the statement is false.

Key Concepts

Sine Function

The sine function, denoted as \( \sin(x) \), is one of the most fundamental trigonometric functions. It is known for its wave-like shape, repeating patterns, and is primarily used to model periodic phenomena. The sine function plots the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle when the angle \( x \) is placed at the origin of a unit circle.

The standard form of the sine function features a smooth and continuous wave that oscillates above and below the horizontal axis. This characteristic waviness is determined by its amplitude, frequency, and phase shift. The basic equation of the sine function is:

• For any real number \( x \), \( y = \sin(x) \)

It's essential to remember that the sine function is symmetric and has a characteristic that makes it "odd", meaning \( \sin(-x) = -\sin(x) \). This plays a crucial role in various applications across physics, engineering, and other fields where wave-like systems are analyzed.

The most recognizable property of the sine function is its period, which we will explore more deeply in the following section.

Period of a Function

The "period" of a function in mathematics refers to the length of one full cycle of the function before it starts repeating itself. For periodic functions like the sine and cosine, the period is the horizontal distance required to complete one full wave.
For example, the basic sine function \( \sin(x) \) has a standard period of \( 2\pi \). This means that the function repeats itself every \( 2\pi \) units along the x-axis. Thus, \( \sin(x + 2\pi) = \sin(x) \).

To determine the period of a modified sine function, such as \( f(t) = \sin(0.05\pi t) \), you'll adjust the period according to the coefficient inside the sine. The equation to find the new period \( T \) is:

• Set up \( k(t + T) = kt + 2\pi \) where \( k \) is the coefficient of \( t \).
• In \( f(t) = \sin(0.05\pi t) \), \( k = 0.05\pi \).
• The modified period \( T \) becomes \( \frac{2\pi}{0.05\pi} = 40 \).

This tells us the function completes one full cycle every 40 units. Understanding periods helps in comprehending how often certain phenomena occur, an essential aspect in fields such as signal processing and harmonic motion analysis.

Trigonometric Functions

Trigonometric functions encompass sine, cosine, and tangent. They are fundamental in understanding relationships in triangles and modeling periodic phenomena. These functions arise naturally in the study of circles and cyclic movements.

The sine function is part of a larger family of trigonometric functions each with their unique characteristics:

• Cosine (\( \cos(x) \)): Similar to sine, but it starts at its maximum value and is also periodic with a period of \( 2\pi \).
• Tangent (\( \tan(x) \)): This function has a period of \( \pi \) and exhibits vertical asymptotes, differing in appearance from the wave-like sine and cosine.

Besides their mathematical beauty, trigonometric functions are vital in calculating angles, distances, and various wave patterns. They play a critical role in architecture, physics, and even sound engineering.

Understanding these functions provides insight into modeling and solving real-world scenarios where waves and cyclic behaviors are present. By mastering trigonometric functions, one gains the ability to delve into fields like acoustics, optics, and quantum physics, carrying the knowledge far beyond just mathematical equations.