By now it is clear that there is a relationship between the angle of a right triangle and its base and height. How can we assign numbers to these relationships? (After all, math people want to measure everything.)
Look at the two sticks below. Even though one is twice as long as the other, both are at a 30 degree angle, so whatever measurement we come up with has to be the same for both sticks.
If you measure the small triangle, you find that, in the case of a thirty degree angle, the height is exactly 50% as long as the hypotenuse. If you measure the big triangle, its height is also 50% as long as the hypotenuse. [They are similar triangles, so this should not surprise you.]
This means we can express the relationship between an angle and the height of a right triangle by dividing the height by the length of the hypotenuse. (If we choose the hypotenuse to be 1 unit in length, the division becomes very easy.)
Likewise, we can express the relationship between an angle and the base of a right triangle by the ratio of the base to the hypotenuse. Click and drag the hypotenuse below to see these ratios in action.