why only $\tan\theta$ is used to get slope and not any other trigonometric functions?

Question

I thought for a while and came up with a reason, which was, maybe it's because the inclination increases with increasing angle; thus, the mathematicians thought that as the angle increases value of $\tan\theta$ increases, so $\tan\theta$ is the apt function for getting a slope of the line. But again it came to my mind that so does $\sin\theta$, starting from 0 to finally 1 for angles $0^\circ$ to $90^\circ$. So why didn't they use it? And why didn't they use $\cot\theta$ instead of $\tan\theta$.

Answer 1

One way to think about why $\tan \theta$ is more appropriate to use as an angle to measure slope is when you drive over a steep grade, e.g. the streets of San Francisco, like Lombard Street.

If you have a hill that has a $30\%$ gradient, it means for every 100 feet you travel, you gain (or lose) 30 feet of elevation. If we wish to measure the angle traveled from the bottom of the hill, we would use $\tan \theta = G$ to find the angle of the slope of this gradient (which would be about 17°). The higher the gradient (i.e. the steeper the hill), the larger $\tan \theta$ will be.