Some people just do not know how to have fun. They think everything has to be useful. Luckily, the sine, cosine, and tangent functions do have a lot of real-world uses.
Surveyors use the tangent function a lot. For example, they can use trigonometry to figure out the distance across rivers.
We first set up a survey post directly across the river from some landmark (like a tree). Then we head downstream a distance that we can measure; in this case, 400 meters. That’s the red horizontal line in the drawing.
Now we take a sighting on the tree from downstream. That’s the black line in the drawing. The surveying instruments will tell us what our sighting angle is. In this case, it’s 31 degrees.
We know from the previous page that the tangent of 31 degrees is equal to the length of the blue line divided by the length of the red line (400 meters). So, if we multiply the tangent of 31 degrees by 400 meters, we’ll get the distance across the river.
The tangent of 31 degrees is about 0.60. That means that the distance across the river is 0.6 times 400 meters, or 240 meters.
Real world alert! Wouldn’t it have been simpler to just tie a rope to the tree, climb into a boat, go across the river, and measure how much rope you trailed out? In this case, it actually might be a good idea to physically cross the river to check the answer from our math. This gives us confidence that trigonometry really works.
After all, sometimes the real world gives us a river that is too difficult to cross or a cliff too dangerous to climb. In these cases the physical approach isn’t possible. We need trigonometry--a mathematical technique we can trust for the answers we need.