Find time period of the function, `y=sin omega t + sin 2omega t + sin 3omega t`

Explanation

To find the time period of the function \( y = \sin(\omega t) + \sin(2\omega t) + \sin(3\omega t) \), we can follow these steps:

### Step 1: Identify Individual Functions
We can express the function as:
- \( y_1 = \sin(\omega t) \)
- \( y_2 = \sin(2\omega t) \)
- \( y_3 = \sin(3\omega t) \)

### Step 2: Determine the Time Period of Each Function
The time period \( T \) of a sine function \( \sin(k\omega t) \) is given by the formula:
\[ T = \frac{2\pi}{k\omega} \]

- For \( y_1 = \sin(\omega t) \):
\[ T_1 = \frac{2\pi}{\omega} \]

- For \( y_2 = \sin(2\omega t) \):
\[ T_2 = \frac{2\pi}{2\omega} = \frac{\pi}{\omega} \]

- For \( y_3 = \sin(3\omega t) \):
\[ T_3 = \frac{2\pi}{3\omega} \]

### Step 3: Find the Least Common Multiple (LCM) of the Time Periods
To find the time period of the combined function \( y \), we need to determine the LCM of \( T_1 \), \( T_2 \), and \( T_3 \):
- \( T_1 = \frac{2\pi}{\omega} \)
- \( T_2 = \frac{\pi}{\omega} \)
- \( T_3 = \frac{2\pi}{3\omega} \)

The LCM of fractions is calculated by taking the LCM of the numerators and dividing it by the GCD of the denominators.

**Numerators:**
- LCM of \( 2\pi, \pi, 2\pi \) is \( 2\pi \).

**Denominators:**
- GCD of \( \omega, \omega, 3\omega \) is \( \omega \).

Thus, the LCM of the time periods is:
\[ T = \frac{\text{LCM of numerators}}{\text{GCD of denominators}} = \frac{2\pi}{\omega} \]

### Final Result
The time period of the function \( y = \sin(\omega t) + \sin(2\omega t) + \sin(3\omega t) \) is:
\[ T = \frac{2\pi}{\omega} \]

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