I was tutoring a student today and they asked a very interesting question while we were looking at the $tan(\theta)=m$ formula for the angle of inclination/the angle with the positive direction of the x-axis and a line with gradient $m$.
We considered the fact that when $m=-1$, the angle made with the positive direction of the x-axis is $$(180-45)^\circ=135^\circ$$
But from here, consider it as an angle from the negative direction of the x-axis: He stated that if $-1$ has angle $45^\circ$, then $m=\frac{-1}{3}$ should have angle equal to $\frac{1}{3}*45^\circ=15^\circ$ from the negative direction of the x-axis which would be $(180-15)^\circ=165^\circ$ from the positive direction of the x-axis.
Why does plugging this same problem into the $tan(\theta)=m$ where $m=\frac{-1}{3}$ give us a different answer?