Find the inclination of the line x-√3y+1=0 with x-axis 

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 \(90^\circ\).

Learn how to calculate the inclination angle of the line x - √3y + 1 = 0 with a detailed step-by-step solution. This problem involves converting the line equation to slope-intercept form, calculating the slope, and using trigonometric functions to find the angle of inclination. Suitable for high school students studying geometry and trigonometry.

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