Find the equation of a line passing through the point `(0,-2)` and makes an angle of `75^(@)` from the positive direction of `X`-axis.

Answer

To find the equation of a line passing through the point (0, -2) and making an angle of 75 degrees with the positive direction of the x-axis, we can follow these steps:

### Step 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:
\[ m = \tan(\theta) \]
Here, \( \theta = 75^\circ \). Therefore:
\[ m = \tan(75^\circ) \]

### Step 2: Use the angle addition formula for tangent
To calculate \( \tan(75^\circ) \), we can use the angle addition formula:
\[ \tan(75^\circ) = \tan(45^\circ + 30^\circ) = \frac{\tan(45^\circ) + \tan(30^\circ)}{1 - \tan(45^\circ) \tan(30^\circ)} \]
Where:
- \( \tan(45^\circ) = 1 \)
- \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \)

Substituting these values:
\[ \tan(75^\circ) = \frac{1 + \frac{1}{\sqrt{3}}}{1 - 1 \cdot \frac{1}{\sqrt{3}}} \]

### Step 3: Simplify the expression
Calculating the numerator and denominator:
\[ = \frac{1 + \frac{1}{\sqrt{3}}}{1 - \frac{1}{\sqrt{3}}} = \frac{\frac{\sqrt{3} + 1}{\sqrt{3}}}{\frac{\sqrt{3} - 1}{\sqrt{3}}} = \frac{\sqrt{3} + 1}{\sqrt{3} - 1} \]

### Step 4: Write the equation of the line
We know the slope \( m \) and a point on the line (0, -2). We can use the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Substituting \( (x_1, y_1) = (0, -2) \):
\[ y - (-2) = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}(x - 0) \]
This simplifies to:
\[ y + 2 = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}x \]

### Step 5: Rearranging the equation
To express the equation in the standard form, we can rearrange it:
\[ y = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}x - 2 \]

### Step 6: Rationalizing the slope
To rationalize the slope \( \frac{\sqrt{3} + 1}{\sqrt{3} - 1} \):
Multiply the numerator and denominator by \( \sqrt{3} + 1 \):
\[ \frac{(\sqrt{3} + 1)(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{3 + 2\sqrt{3} + 1}{3 - 1} = \frac{4 + 2\sqrt{3}}{2} = 2 + \sqrt{3} \]

### Final equation
Thus, the equation of the line can be written as:
\[ y + 2 = (2 + \sqrt{3})x \]
or
\[ y = (2 + \sqrt{3})x - 2 \]