Ask most people to find the area of a triangle, and they'll reach for "base × height ÷ 2." But to use that formula, you need the height measured perpendicular to the base — and actually drawing the shape and measuring that height yourself is more of a hassle than it sounds. Trig ratios make that hassle disappear entirely. Know just two sides a, b and the included angle C between them, and you can calculate it directly: Area = (1/2)×a×b×sinC.
Why does this work? Call the perpendicular height dropped from base a to the opposite vertex h. In the right triangle formed by side b, height h, and angle C, the relationship sinC = h/b holds. So h = b×sinC, and substituting this into the original formula (1/2)×a×h gives (1/2)×a×b×sinC. In the end, the trig ratio formula is nothing more than "computing the height from one side and one angle, instead of measuring the base and height directly." In fact, no matter which angle you pick as the included angle, the area always comes out the same — which is further proof that this formula is really the same area formula in disguise.
Trig ratios aren't just for area — they're also used to find distances you can't measure directly. A classic example is measuring the height of a building or a tree. Stand some distance from the base of a building, and use a protractor or clinometer to measure the angle between the horizontal and your line of sight to the top of the building (this angle is called the angle of elevation). With just the horizontal distance x from your observation point to the building and the angle of elevation θ, you can find the height. In the right triangle formed by the observer, the base of the building, and the top of the building, tanθ = height/x, so height h = x × tanθ follows.
But what if you don't know exactly how close you are to the building — that is, you don't know the horizontal distance x? In that case, use the method of measuring the angle of elevation from two different spots. Measure the angle of elevation from a point A far from the building, and from a point B some fixed distance closer to the building, and you get two equations in two unknowns (the building's height, and the distance from B to the building). Solve these two equations together, and you can find the exact height without ever knowing the distance. This very principle (triangulation) has long been used to measure the height of mountains and even the distance to stars.
In short, trig ratios are a tool for calculating "things you can't measure with a ruler" using only a handful of angles and distances. On the activity page, try changing the sides and included angle with sliders to see how the area changes, and change the distance and angle of elevation to see how a building's height is calculated.