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Find the angle between the lines whose s...

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

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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 \( 60^\circ \).
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