Find the period of following function
` sin x + cos 2x `

Answer

To find the period of the function \( f(x) = \sin x + \cos 2x \), we will follow these steps:

### Step 1: Identify the periods of the individual functions
The period of \( \sin x \) is \( 2\pi \) and the period of \( \cos 2x \) can be calculated using the formula for the period of cosine functions, which is given by \( \frac{2\pi}{n} \) where \( n \) is the coefficient of \( x \). Here, \( n = 2 \).

**Calculation:**
\[ \text{Period of } \cos 2x = \frac{2\pi}{2} = \pi \]

### Step 2: List the periods
Now we have:
- Period of \( \sin x = 2\pi \)
- Period of \( \cos 2x = \pi \)

### Step 3: Find the least common multiple (LCM)
To find the period of the sum of these functions, we need to find the least common multiple (LCM) of their periods.

**Calculation:**
- The periods are \( 2\pi \) and \( \pi \).
- The LCM of \( 2\pi \) and \( \pi \) is \( 2\pi \).

### Step 4: Conclusion
Thus, the period of the function \( f(x) = \sin x + \cos 2x \) is:
\[ \text{Period} = 2\pi \]