Does the sign of the numerator matter in the "arctan-angle-line" formula?

Question

So the simplified formula for finding the acute angle between two intersecting lines is: tan-¹ (m1-m2/1+m1m2) But what if m2 is larger than m1 and we get a negative numerator? Does the formula still hold?

Answer 1

The right formula is $$\alpha=\arctan\left|\frac{m_1-m_2}{1+m_1m_2}\right|$$

Let $l_1$ and $l_2$ be two straight lines and $\alpha$ be a measure of the angle between $l_1$ and $l_2$.

Thus, if $l_1||l_2$ then $\alpha=0^{\circ}$;

if $l_1\perp l_2$ then $\alpha=90^{\circ}$;

otherwise, $\alpha$ it's a measure of the small angle between four angles,

which which we get by these lines.