Question

Period of \(|\sin t|\) is

Short answer

The period of \( |\sin t| \) is \( \pi \).

Step-by-step solution

Understand the Basic Function

The sine function, \( \sin t \), is a periodic function with a period of \( 2\pi \). It means that \( \sin(t + 2\pi) = \sin t \) for any value of \( t \).

Consider the Absolute Value

By taking the absolute value, \( |\sin t| \) becomes a non-negative function. The effect of the absolute value is that it reflects the negative parts of \( \sin t \) above the horizontal axis, creating a 'V' shape at points where \( \sin t \) is zero.

Determine the New Period

For \( |\sin t| \), the pattern repeats every \( \pi \) because for half of the \( 2\pi \) interval, \( \sin t \) is non-negative, and for the other half, it mirrors itself due to the absolute value. Thus, \( |\sin t| \) fulfills \( |\sin(t + \pi)| = |\sin t| \).

Confirm the Periodicity

To confirm, observe that within one full cycle from 0 to \( \pi \), the function \( |\sin t| \) starts at 0, reaches 1 at \( \frac{\pi}{2} \), and returns to 0 at \( \pi \). Afterward, it exhibits the same behavior from \( \pi \) to \( 2\pi \).

Key concepts

Periodic Functions

Periodic functions are functions that repeat their values at regular intervals or periods. This means that if you observe the function for a specified interval, the behavior of the function will repeat itself in the next interval of the same length. For example, in the context of the trigonometric functions like the sine or cosine, these intervals are a fundamental concept since they define how these functions behave. Without understanding periodicity, predicting or analyzing these functions would be challenging.

Periodic functions have some characteristic features:

• They have a fundamental period, which is the length of one complete cycle of the function.
• The function values repeat every interval of one period.
• Graphically, you can see that the waves or patterns of the function recurrences match over their periods.

In mathematics and the real world, periodic functions are present everywhere, like sound waves or the rotation of a wheel. Understanding this concept is essential, especially when studying trigonometric functions.

Sine Function

The sine function, denoted as \( \sin t \), is one of the most basic and well-known trigonometric functions. Its graph is a smooth, continuous wave that cycles between -1 and 1, and repeats itself every \(2\pi\) radians. This characteristic makes the sine function a periodic function.

Here are some key aspects of the sine function:

• The period of \( \sin t \) is \( 2\pi \), which means every \( 2\pi \) radians, the function starts repeating its values from the beginning.
• It is an odd function, meaning that \( \sin(-t) = -\sin(t) \).
• It reaches its maximum value of 1 at \( \frac{\pi}{2} + 2k\pi \) and its minimum value of -1 at \( \frac{3\pi}{2} + 2k\pi \) for any integer \( k \).

Understanding the sine function's periodic behavior is essential in solving various mathematical problems involving oscillations and waves.

Absolute Value Function

The absolute value function, denoted as \(|f(x)|\), takes any real-valued function and converts all its values to non-negative outcomes. By applying absolute value to a function, you effectively "flip" any negative parts of its graph above the x-axis, creating a reflection.

• For every negative value \(x\), \(|x| = -x\).
• For every non-negative value \(x\), \(|x| = x\).
• This operation modifies functions, changing their behavior while also affecting their graph.

In terms of our example \(|\sin t|\), the absolute value reflects the graph of \( \sin t \) above the x-axis wherever \(\sin t\) is negative, resulting in a non-negative wave-like pattern that maintains much of the original periodic structure.

Function Periodicity

Function periodicity refers to a function's tendency to repeat its values at regular intervals known as "periods." For periodic functions, once the period is determined, the function's behavior can be predicted indefinitely in both directions along the x-axis.

For \(|\sin t|\), determining the period involves understanding both the sine function and the modifications by the absolute value:

• The regular sine function \( \sin t \) has a period of \( 2\pi \).
• Applying the absolute value changes the period to \( \pi \) because \(|\sin t|\) creates a symmetry about each \(\pi\), essentially halving the original sine period.
• This results in \(|\sin(t + \pi)| = |\sin t|\) where the wave pattern repeats.

Recognizing the periodicity in \(|\sin t|\) helps in understanding how the function behaves over intervals and applies to problem-solving in trigonometry and other applications requiring prediction of cyclic behaviors.