The inclination (or angle of inclination) of a line with the $x$-axis is the angle $\theta$ that the line makes with the positive direction of the $x$-axis. This angle can be calculated using the slope of the line.

Given the equation of the line:

$x - \sqrt{3}y + 1 = 0$

we can rewrite it in slope-intercept form:

$y = mx + c$

where $m$ is the slope of the line.

Step 1: Convert to Slope-Intercept Form

First, rearrange the given equation:

$\sqrt{3}y = x + 1$

$y = \frac{1}{\sqrt{3}}x + \frac{1}{\sqrt{3}}$

Here, the slope $m = \frac{1}{\sqrt{3}}$.

Step 2: Calculate the Inclination Angle

The angle of inclination $\theta$ is given by:

$\theta = \tan^{-1}(m)$

Substitute the value of $m$:

$\theta = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right)$

Step 3: Find the Value of $\theta$

$\theta = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = 30^\circ$

Conclusion

The inclination of the line $x - \sqrt{3}y + 1 = 0$ with the $x$-axis is 30 degrees.

Would you like more details on any step or have any questions?

Here are 5 related questions that might interest you:

1. How do you find the inclination of a line given two points on the line?
2. What is the relationship between the slope and angle of inclination of a line?
3. How do you find the angle between two lines?
4. What happens to the angle of inclination if the slope is negative?
5. Can the inclination of a line be greater than $90^\circ$?

Tip: The tangent function is periodic, and its inverse (arctan) only gives angles between $-90^\circ$ and $90^\circ$.