Find the equation of a line passing through the origin and making an angle of `120^(@)` with the positive direction of the x-axis.

Answer

To find the equation of a line passing through the origin and making an angle of \(120^\circ\) with the positive direction of the x-axis, we can follow these steps:

### Step 1: Identify the angle
The angle given is \(120^\circ\). This angle is measured counterclockwise from the positive x-axis.

**Hint:** Remember that angles in standard position are measured from the positive x-axis.

### Step 2: Calculate the slope of the line
The slope \(m\) of a line that makes an angle \(\theta\) with the positive x-axis can be calculated using the tangent function:
\[ m = \tan(\theta) \]
For \(\theta = 120^\circ\):
\[ m = \tan(120^\circ) \]

**Hint:** Use the identity \(\tan(180^\circ - \theta) = -\tan(\theta)\) to simplify calculations.

### Step 3: Simplify the slope
Since \(120^\circ\) is in the second quadrant, we can express it as:
\[ \tan(120^\circ) = \tan(180^\circ - 60^\circ) = -\tan(60^\circ) \]
We know that \(\tan(60^\circ) = \sqrt{3}\), hence:
\[ \tan(120^\circ) = -\sqrt{3} \]
Thus, the slope \(m\) is:
\[ m = -\sqrt{3} \]

**Hint:** Remember that the slope of a line in the second quadrant is negative.

### Step 4: Use the point-slope form of the equation of a line
The point-slope form of the equation of a line is given by:
\[ y - y_1 = m(x - x_1) \]
Since the line passes through the origin \((0, 0)\), we have \(x_1 = 0\) and \(y_1 = 0\):
\[ y - 0 = -\sqrt{3}(x - 0) \]
This simplifies to:
\[ y = -\sqrt{3}x \]

**Hint:** The point-slope form is useful when you know a point on the line and the slope.

### Step 5: Rearranging the equation
To express the equation in standard form, we can rearrange it:
\[ \sqrt{3}x + y = 0 \]

**Hint:** Standard form of a line is often written as \(Ax + By + C = 0\).

### Final Answer
The equation of the line passing through the origin and making an angle of \(120^\circ\) with the positive direction of the x-axis is:
\[ \sqrt{3}x + y = 0 \]