The slope of a line that makes an angle of measure `30^(@)` with Y-axis is

Options

A. `sqrt3`
B. `-sqrt3`
C. `pm sqrt3`
D. `pm (1)/(sqrt3)`

Answer

To find the slope of a line that makes an angle of \(30^\circ\) with the Y-axis, we can follow these steps:

### Step 1: Understand the relationship between the angle and the slope
The slope \(m\) of a line can be calculated using the tangent of the angle it makes with the X-axis. However, since we are given the angle with the Y-axis, we need to adjust our approach.

### Step 2: Calculate the angle with the X-axis
If the line makes an angle of \(30^\circ\) with the Y-axis, the angle \(\theta\) it makes with the X-axis can be found using:
\[ \theta = 90^\circ - 30^\circ = 60^\circ \]

### Step 3: Calculate the slope using the tangent function
The slope \(m\) of the line is given by:
\[ m = \tan(\theta) \]
Substituting the value of \(\theta\):
\[ m = \tan(60^\circ) \]

### Step 4: Find the value of \(\tan(60^\circ)\)
From trigonometric values, we know:
\[ \tan(60^\circ) = \sqrt{3} \]
Thus,
\[ m = \sqrt{3} \]

### Step 5: Consider the negative slope scenario
Since the angle can also be measured in the opposite direction (making an angle of \(30^\circ\) with the Y-axis in the negative direction), we need to consider:
\[ \theta = -60^\circ \]
Then, the slope for this angle would be:
\[ m = \tan(-60^\circ) \]
Using the property of tangent:
\[ \tan(-\theta) = -\tan(\theta) \]
We find:
\[ m = -\tan(60^\circ) = -\sqrt{3} \]

### Final Result
Thus, the slope of the line that makes an angle of \(30^\circ\) with the Y-axis can be either:
\[ m = \sqrt{3} \quad \text{or} \quad m = -\sqrt{3} \]