Understanding the Period of Trigonometric Functions

The question asks for the period of the function \(f(x) = \sin x\). The period of a function is the smallest positive value \(T\) such that \(f(x + T) = f(x)\) for all values of \(x\) in the domain of the function. In simpler terms, it's the length of the smallest interval over which the function's graph repeats itself.

Analyzing the Sine Function \(f(x) = \sin x\)

Let's consider the properties of the sine function. The graph of \(y = \sin x\) starts at 0 at \(x=0\), increases to 1 at \(x=\pi/2\), decreases to 0 at \(x=\pi\), decreases to -1 at \(x=3\pi/2\), and increases back to 0 at \(x=2\pi\). After \(x=2\pi\), the pattern repeats.

We need to find the smallest positive value \(T\) such that \(\sin(x + T) = \sin x\).

We know from trigonometric identities that:

• \(\sin(x + 2\pi) = \sin x\)
• \(\sin(x + 4\pi) = \sin x\)
• \(\sin(x + 6\pi) = \sin x\)

and so on. The values \(2\pi, 4\pi, 6\pi, \dots\) are all values of \(T\) for which the property holds.

However, the period is defined as the smallest positive value of \(T\). By examining the graph of \(\sin x\) or understanding its definition in terms of the unit circle, we see that the function completes one full cycle (from 0 up to 1, down to -1, and back to 0) exactly over an interval of length \(2\pi\). For example, from \(x=0\) to \(x=2\pi\), or from \(x=\pi/2\) to \(x=5\pi/2\).

If we take any value \(T\) smaller than \(2\pi\) (but positive), there will be some \(x\) for which \(\sin(x+T) \neq \sin x\). For instance, if \(T=\pi\), \(\sin(x+\pi) = -\sin x\), which is not equal to \(\sin x\) for all \(x\).

Therefore, the smallest positive value \(T\) for which \(\sin(x+T) = \sin x\) is \(2\pi\).

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

Comparing with Options

Let's look at the given options:

• Option 1: \(\pi/4\)
• Option 2: \(\pi/2\)
• Option 3: \(\pi\)
• Option 4: \(2\pi\)

Our calculated period for \(f(x) = \sin x\) is \(2\pi\), which matches Option 4.

Summary of Period for Basic Sine Function

• Function: \(f(x) = \sin x\)
• Property: \(f(x + T) = f(x)\)
• Smallest positive \(T\): \(2\pi\)
• Period: \(2\pi\)

Revision Table: Key Trigonometric Periods

Additional Information on Trigonometric Periods

The period of a trigonometric function determines how often its graph repeats. For \(f(x) = \sin x\), the graph repeats every \(2\pi\) units along the x-axis. The coefficient of \(x\) inside the sine function affects the period. For a function like \(g(x) = \sin(bx)\), the period is \(2\pi/|b|\). In our question, \(f(x) = \sin x\), which can be thought of as \(\sin(1 \cdot x)\). Here, \(b=1\), so the period is \(2\pi/|1| = 2\pi\).

Understanding the period is crucial for graphing trigonometric functions and solving trigonometric equations, as it helps identify all possible solutions within repeating intervals.