Question

derive the formula for The angle between two straight lines with slopes m1 m2.hence find the acute angle between the lines .​

Answer 1

Answer:

m1=m2 is the required formula

Answer 2

$Y^{\circ} = \tan^{-1}[\pm \frac{m_{1} - m_{2}}{1 + m_{1}m_{2}}]$

Step-by-step explanation:

Let AB and CD are the two straight lines that make angles to the positive x-axis X° and Z°.

So, as per given conditions, $m_{1} = \tan X^{\circ}$ and $m_{2} = \tan Z^{\circ}$

Now, from the closed triangle shown in the diagram attached,

X° = Y° + Z°

⇒ Y° = X° - Z°

⇒ $\tan Y^{\circ} = \tan (X^{\circ} - Z^{\circ})$

⇒ $\tan Y^{\circ} = \frac{\tan X^{\circ} - \tan Z^{\circ}}{1 + \tan X^{\circ}\tan Z^{\circ}}$

⇒ $\tan Y^{\circ} = \frac{m_{1} - m_{2}}{1 + m_{1}m_{2}}$

⇒ $Y^{\circ} = \tan^{-1}[\frac{m_{1} - m_{2}}{1 + m_{1}m_{2}}]$

So, this is the expression for the acute angle between the line AB and CD.

Now, for the obtuse angle (P°) we can write the expression as $P^{\circ} = \tan^{-1}[- \frac{m_{1} - m_{2}}{1 + m_{1}m_{2}}]$ (Answer)