Find the angle between the lines whose slopes are `(2-sqrt3) ` and `2+sqrt3)` .

Answer

To find the angle between the lines whose slopes are \( m_1 = 2 - \sqrt{3} \) and \( m_2 = 2 + \sqrt{3} \), we can use the formula for the tangent of the angle \( \theta \) between two lines:

\[ \tan \theta = \frac{m_2 - m_1}{1 + m_1 m_2} \]

### Step 1: Identify the slopes
We have:
- \( m_1 = 2 - \sqrt{3} \)
- \( m_2 = 2 + \sqrt{3} \)

### Step 2: Calculate \( m_2 - m_1 \)
\[ m_2 - m_1 = (2 + \sqrt{3}) - (2 - \sqrt{3}) = 2 + \sqrt{3} - 2 + \sqrt{3} = 2\sqrt{3} \]

### Step 3: Calculate \( m_1 m_2 \)
\[ m_1 m_2 = (2 - \sqrt{3})(2 + \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]

### Step 4: Calculate \( 1 + m_1 m_2 \)
\[ 1 + m_1 m_2 = 1 + 1 = 2 \]

### Step 5: Substitute into the tangent formula
Now we can substitute \( m_2 - m_1 \) and \( 1 + m_1 m_2 \) into the tangent formula:
\[ \tan \theta = \frac{2\sqrt{3}}{2} = \sqrt{3} \]

### Step 6: Find \( \theta \)
To find \( \theta \), we take the arctangent:
\[ \theta = \tan^{-1}(\sqrt{3}) \]
From trigonometric values, we know that:
\[ \theta = 60^\circ \]

Thus, the angle between the lines is \( 60^\circ \).