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To determine the equation of the line that passes through the point \((0, 4)\) and makes an angle of \(30^\circ\) with the positive direction of the x-axis, we can use the following steps:

### Step-by-Step Solution

1. **Find the slope of the line:**

   The slope \(m\) of a line that makes an angle \(\theta\) with the positive x-axis is given by:

   \[
   m = \tan(\theta)
   \]

   Given that \(\theta = 30^\circ\):

   \[
   m = \tan(30^\circ) = \frac{1}{\sqrt{3}}
   \]

2. **Use the point-slope form of the line equation:**

   The point-slope form of a line passing through a point \((x_1, y_1)\) with slope \(m\) is:

   \[
   y - y_1 = m(x - x_1)
   \]

   Substituting the point \((0, 4)\) and the slope \(m = \frac{1}{\sqrt{3}}\):

   \[
   y - 4 = \frac{1}{\sqrt{3}}(x - 0)
   \]

   Simplifying:

   \[
   y = \frac{1}{\sqrt{3}}x + 4
   \]

3. **Convert to standard form (if needed):**

   The equation of the line in slope-intercept form is:

   \[
   y = \frac{1}{\sqrt{3}}x + 4
   \]

   This is the equation of the line passing through the point \((0, 4)\) and making an angle of \(30^\circ\) with the positive x-axis.

### Final Answer

The equation of the line is:

\[
y = \frac{1}{\sqrt{3}}x + 4
\]

Would you like more details or have any other questions?

### Related Questions:
1. What is the general form of the equation for this line?
2. How would the equation change if the line passed through a different point, say \((1, 3)\)?
3. What is the y-intercept and x-intercept of this line?
4. How do you find the equation of a line if it makes a different angle, like \(45^\circ\) with the x-axis?
5. Can you determine the distance between the origin and the line?

### Tip:
When given an angle with the x-axis, always use the tangent of that angle to determine the slope of the line.

A line passes through the point (0, 4) and makes an angle of 30° with the positive direction of the x-axis. Determine the equation of the line.

Learn how to find the equation of a line passing through the point (0, 4) and making an angle of 30° with the x-axis. This problem involves finding the slope using trigonometry and applying the point-slope formula to determine the equation.

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