A paricle starting from the origin (0,0) moves in a straight line in `(x,y)` plane. Its coordinates at a later time are `(sqrt(3),3)`. The path of the particle makes with the x-axis an angle of

Options

A. `30^(@)`
B. `45^(@)`
C. `60^(@)`
D. `0^(@)`

Explanation

To solve the problem of finding the angle that the path of the particle makes with the x-axis, we can follow these steps:

### Step 1: Identify the coordinates of the particle
The particle starts from the origin (0,0) and moves to the coordinates \((\sqrt{3}, 3)\).

### Step 2: Determine the change in coordinates
The change in the x-coordinate is \(\sqrt{3} - 0 = \sqrt{3}\) and the change in the y-coordinate is \(3 - 0 = 3\).

### Step 3: Use the tangent function to find the angle
The angle \(\theta\) that the path makes with the x-axis can be found using the tangent function:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{y}{x} = \frac{3}{\sqrt{3}} \]

### Step 4: Simplify the expression
Now, simplify \(\frac{3}{\sqrt{3}}\):
\[ \tan(\theta) = \frac{3}{\sqrt{3}} = \sqrt{3} \]

### Step 5: Find the angle using the inverse tangent function
To find \(\theta\), we take the arctangent (inverse tangent) of \(\sqrt{3}\):
\[ \theta = \tan^{-1}(\sqrt{3}) \]

### Step 6: Determine the angle in degrees
From trigonometric values, we know that:
\[ \tan(60^\circ) = \sqrt{3} \]
Thus,
\[ \theta = 60^\circ \]

### Conclusion
The angle that the path of the particle makes with the x-axis is \(60^\circ\).

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