To find the equation of the line that makes an angle of $120^\circ$ with the X-axis and passes through the point $(2, 3)$, we can proceed as follows:

Step 1: Find the slope of the line

The slope $m$ of a line that makes an angle $\theta$ with the X-axis is given by: $m = \tan(\theta)$ Here, $\theta = 120^\circ$. So, $m = \tan(120^\circ)$ Using the tangent of $120^\circ$: $\tan(120^\circ) = \tan(180^\circ - 60^\circ) = -\tan(60^\circ)$ Since $\tan(60^\circ) = \sqrt{3}$, we have: $m = -\sqrt{3}$

Step 2: Use the point-slope form of the equation of a line

The point-slope form of the equation of a line is: $y - y_1 = m(x - x_1)$ where $(x_1, y_1)$ is a point on the line and $m$ is the slope. Here, $(x_1, y_1) = (2, 3)$ and $m = -\sqrt{3}$.

Substitute the values: $y - 3 = -\sqrt{3}(x - 2)$

Step 3: Simplify the equation

Expand and simplify the equation: $y - 3 = -\sqrt{3}x + 2\sqrt{3}$ $y = -\sqrt{3}x + 2\sqrt{3} + 3$

Thus, the equation of the line is: $y = -\sqrt{3}x + (2\sqrt{3} + 3)$

Let me know if you need further clarifications.

Related Questions:

1. What is the equation of a line passing through the origin and making an angle of $45^\circ$ with the X-axis?
2. How do you derive the general form of the equation of a line given its slope and a point?
3. What happens to the slope if the angle made with the X-axis is $90^\circ$?
4. How do you find the equation of a line perpendicular to the line obtained here?
5. How can we check if two lines are parallel based on their equations?

Tip:

Always remember that the slope of a line is negative when the angle with the X-axis is between $90^\circ$ and $180^\circ$.