The position of a particle moving along x-axis varies with time t according to equation `x=sqrt(3) sinomegat-cosomegat` where `omega` is constants. Find the region in which the particle is confined.

Options

A. `-2lt=xlt=2`
B. `-3lt=xlt=3`
C. `-sqrt(2)lt=xlt=sqrt(2)`
D. `-1lt=xlt=1`

Explanation

To find the region in which the particle is confined, we start with the given equation of motion:

\[ x = \sqrt{3} \sin(\omega t) - \cos(\omega t) \]

### Step 1: Rewrite the equation
We can rewrite the equation by factoring out a common term. We will express the equation in a form that allows us to use the sine subtraction formula.

### Step 2: Factor the expression
We can express the equation as follows:

\[ x = \sqrt{3} \sin(\omega t) - \cos(\omega t) \]

Notice that we can factor out a 2 from the terms:

\[ x = 2 \left( \frac{\sqrt{3}}{2} \sin(\omega t) - \frac{1}{2} \cos(\omega t) \right) \]

### Step 3: Identify trigonometric identities
Recognizing the coefficients, we see that:

- \(\frac{\sqrt{3}}{2} = \sin(60^\circ)\)
- \(\frac{1}{2} = \cos(60^\circ)\)

Thus, we can rewrite the expression using the sine subtraction formula:

\[ x = 2 \left( \sin(\omega t) \cos(60^\circ) - \cos(\omega t) \sin(60^\circ) \right) \]

This can be expressed as:

\[ x = 2 \sin\left(\omega t - 60^\circ\right) \]

### Step 4: Determine the range of the sine function
The sine function oscillates between -1 and 1. Therefore, we can write:

\[ -1 \leq \sin\left(\omega t - 60^\circ\right) \leq 1 \]

### Step 5: Scale the range
Multiplying the entire inequality by 2 gives:

\[ -2 \leq 2 \sin\left(\omega t - 60^\circ\right) \leq 2 \]

Thus, we have:

\[ -2 \leq x \leq 2 \]

### Conclusion
The particle is confined in the region:

\[ x \in [-2, 2] \]