What is the period of the function f(x)=sin x?

Options

A. `pi//4`
B. `pi//2`
C. `pi`
D. `2pi`

Answer

To find the period of the function \( f(x) = \sin x \), we can follow these steps:

### Step 1: Understand the definition of a periodic function
A function \( f(x) \) is said to be periodic if there exists a positive constant \( T \) such that:
\[ f(x) = f(x + T) \]
for all \( x \).

### Step 2: Apply the definition to the sine function
For the sine function, we need to find a value \( T \) such that:
\[ \sin x = \sin(x + T) \]

### Step 3: Use the properties of the sine function
The sine function has a known property that:
\[ \sin(x + 2\pi) = \sin x \]
This means that the sine function repeats its values every \( 2\pi \).

### Step 4: Identify the smallest positive period
The smallest positive value of \( T \) that satisfies the periodic condition is known as the fundamental period. From the property mentioned, we see that:
\[ T = 2\pi \]
is the period of the sine function.

### Step 5: Conclusion
Thus, the period of the function \( f(x) = \sin x \) is:
\[ \text{Period} = 2\pi \]