Use Of The Tangent Function?

It's been a while since I took a trigonometry class. But ever since I did, one thing has vexed me.

The Sine function is pretty straightforward on where it comes up. It actually comes up quite frequently in science and nature. In fact, most people have actually long been exposed to a Sine function before they even knew what it was.

But what about the Tangent function? Where and when does that come up in nature and science? I've never seen it anywhere else, that is anywhere else but a trigonometry class.

Here, you can see what I am talking about:

 

If you are trying to make a perfect 90 degree angle you can never really do so. You can only make two surfaces almost 90, and that corresponds to the tangent function. As you get closer and closer to 90 degrees the tangent is just shooting up towards infinity. Tan(85)=11.4 Tan(88)=28.8 Tan(89)=57.3 Tan(89.9)=573 Tan(89.99)=5,730 ... So you can see it is impossible to make two things perfectly 90 degrees from each other. The closer you get to 90 the larger the tan is, but when you get close to 90 the tan is wildly large.

Another thing about the tan is that it shows up in spirals made of right triangles that are built in the opposite way from a Spiral of Theodorus. In a spiral of Theodorus (Spiral of Theodorus - Wikipedia) you're stacking rights with one leg as the hypotenuse of another. I'm talking about the other arrangement where you've got leg on leg, going round getting larger and larger very quickly. In that case the growth of the triangles is proportional to the tangent. Its a multiple.

 

Correction: its not proportional to the tangent. The sizes of succeeding triangles is proportional to powers of the tangents. Or if you're going the direction of shrinking triangles they are proportional to powers of cotangents.

 

The tangent probably shows up in more practical situations than sin or cos. Think of measuring to build a sloping roof. You have a certain amount of height from the rafters to the peak, and a certain horizontal span--but what's the actual angle of the slope? Well, it's the angle with a tangent formed by height divided by span (most likely half the span, unless it's a lean-to roof). Or you see a road sign saying "20% grade" and you wonder what that is as an angle. Again it's the angle which has 0.2 as its tangent (and it's 11.3 degrees).