I was taught that the tangent of the angle between two lines having slopes $m_1$ and $m_2$ is given by the formula modulus of $\frac{m_1-m_2}{1+m_1m_2}$. We can then use $\tan^{-1}$ to find the angle.
However, some angles have negative tangent values, which will not be obtained by this formula which uses modulus. But shouldn't these angles also exist between two lines?
On
My interpretation of the vertical bars in the formula ($\tan \theta = |\dfrac {m_1 – m_2}{1 + m_1m_2}|$) is NOT modulus but absolute value instead.
This means $\tan \theta = + \dfrac {m_1 – m_2}{1 + m_1m_2}$ or $\tan \theta =–\dfrac {m_1 – m_2}{1 + m_1m_2}$.
If one value does not give an acute value of $\theta$, the other will.
Before you use the formula, you should determine what type of angle you are looking for, specifically, acute or obtuse - when two lines intersect, two pairs of identical angles are formed. To specify which angle you are targeting, use your formula, with $m_{2}$ being the angle's starting line. If you get a negative output after taking inverse tangent, just take the positive of the answer. This results from the fact that inverse tangent is an odd function; specifically, $\arctan(-\theta) = -\arctan(\theta).$ Hope this helps!