Real world tangent functions

Question

I am a high school math teacher and one of my students asked me for examples of real world tangent functions. Not using tangent to find a side length but a relationship that can be represented by a tangent function. I can not think of any or find any, please help!

Answer 1

One example of applying tangent functions to solve a real world problem is:-

Find the gradient and the actual length of a path represented as x cm (a known measurable quantity) in the 1 : n (a known given quantity) scaled contour map.

Answer 2

Rate of altitude change for an aircraft = groundspeed $\times$ tan(flight path angle). I work with this sort of thing all the time in "real life."

Now suppose the aircraft flies a rhumb line (path that always travels along the same true course, that is, always at the same angle to the left or right of the direction to the north pole) from a point at latitude $lat_1$ and longitude $lon_1$ to a point at latitude $lat_2$ and longitude $lon_2.$ The true course is then $$\psi = \mbox{atan$2$}\left(lon_1-lon_2, \ln\left(\frac{\tan\left(\frac{lat_2}{2}+\frac\pi4\right)} {\tan\left(\frac{lat_1}{2}+\frac\pi4\right)}\right)\right).$$ (In practice there are a couple of additional wrinkles because longitudes suddenly jump from $179$W to $179$E and aviators like their course to be in the range $0$ to $360$ whereas atan$2$ returns values from $-\pi$ to $\pi$, but we continue to use tangents much as in this simplified formula.)

If an aircraft flies in a level turn with a constant bank angle of $\theta$ at a constant groundspeed $V$ (most likely if there is no wind), then the radius of the turn is $$R = \frac{V^2}{g\tan\theta}.$$ This formula also governs the ideal relationship between the bank angle of a turn in a racetrack, the radius of the turn, and the speed you travel in the turn.