Problem 11 - Algebra 2 and Trigonometry

Question

Find the period of each function. \(y=\sin x\)

Step-by-Step Solution

Answer

The period of the function \(y = \sin x\) is \(2\pi\).

Step 1: Understand the Sine Function

The standard sine function is denoted as \(y = \sin x\). The graph of this function is a continuous wave that repeats itself at regular intervals.

Step 2: Identify the Standard Period of Sine

The period of a function is the smallest interval after which the function repeats itself. For \( y = \sin x \), the standard period is \(2\pi\). This means the sine wave starts repeating after an interval of \(2\pi\) units on the x-axis.

Step 3: Conclusion

Since the given function \(y = \sin x\) matches the standard sine function, the period of this function is \(2\pi\).

Key Concepts

Sine Function

The sine function is one of the most fundamental trigonometric functions, and it is represented as \( y = \, \sin x \). It graphically displays a smooth, continuous wave that extends in both positive and negative directions along the x-axis. The wave-like graph crosses zero at regular intervals, which shows its symmetry and balance.
When examined over one complete cycle, the sine wave starts at zero, peaks at one, returns to zero, dips to negative one, and finally returns to zero again. This oscillatory nature is deeply rooted in how the sine function is derived from the unit circle in trigonometry.

• The maximum value of \( \, \sin x \) is 1.
• The minimum value is -1.
• The function is continuous and smooth, without any breaks on its graph.

Understanding the sine function and its graph is crucial in various applications, including sound waves, light waves, and other periodic phenomena.

Periodicity

Periodicity refers to the feature of certain functions, including the sine function, to repeat values at equal intervals. In mathematical terms, the period of a function is the smallest positive number \( T \) for which the equation \( f(x + T) = f(x) \) holds true for all values of \( x \).
For the standard sine function \( y = \, \sin x \), the period is \(2\pi\), which means every \(2\pi\) interval along the x-axis, the sine wave repeats its pattern.
It's important to note:

• The entire wave pattern from one cycle is contained in this interval of \(2\pi\).
• The periodic nature of the sine function allows us to predict its behavior beyond the \(2\pi\) span by simply repeating the wave pattern.
• Periodicity helps simplify complex trigonometric equations and integrals by using this recurring property.

Understanding periodicity is key to many areas like signal processing, where it's imperative to understand how waves recur and interact over time.

Wave Behavior

Wave behavior encompasses the way sinusoidal waves, like the sine wave, move and interact. It is a critical concept in physics and engineering, particularly in studying oscillations, sound, and electromagnetic waves.
The sine wave's behavior is characterized by its repeating peaks and troughs, smooth continuity, and consistent periodicity. When observing wave behavior of \( y = \, \sin x \):

• The wave crosses the x-axis at every half-period \( \pi \), showing points called nodes.
• The highest point, or crest, in each cycle corresponds to the maximum value of the function (1), while the lowest point, or trough, corresponds to its minimum (-1).
• This behavior simulates natural phenomena like tides, sound, and light, which naturally follow repetitive cycles.

By understanding wave behavior, we can apply the principles of trigonometry to analyze and predict the behavior of real-world wave patterns. This has applications not just in mathematics, but also in fields like acoustics, radio transmissions, and even quantum mechanics.