I was taught that the tangent of the angle between two lines having slopes $m_1$ and $m_2$ has modulus of $\frac{m_1-m_2}{1+m_1m_2}$.

Question

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?

You might consider how many angles are created when two lines intersect. — euler1944, Jan 10 '16 at 05:00
Negative angles might be that they are in clockwise direction thats why we use mod also intersection creates $4$ angles. — Archis Welankar, Jan 10 '16 at 05:21

score 1 · Answer 1 · answered Jan 10 '16 at 05:31

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!

score 0 · Answer 2 · answered Jan 10 '16 at 05:38

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.

I was taught that the tangent of the angle between two lines having slopes $m_1$ and $m_2$ has modulus of $\frac{m_1-m_2}{1+m_1m_2}$.

2 Answers2