If you are viewing a right triangle from the viewpoint of one of the smaller angles, there are six trigonometric functions that relate to the angle. For example, the length of the side opposite the angle divided by the hypotenuse of the triangle is called the sine of the angle, written Sin A, where A is the angle. If you know one side and one angle, you can compute the other sides. For example, if the angle is 30 degrees and you know that the hypotenuse is 100 meters, Sin 30 = 0.5 = X/100; so the far side is 50 meters.
Using all six functions and a little algebra, trigonometry can be used to figure out the width of rivers and height of cliffs by taking a couple of angles and measuring off a distance on some convenient place. For example, the cotangent of A, written Cot A, is the side along the base of the angle divided by the side opposite the angle. If you measure off a distance "d" that is along a line perpendicular to a cliff and the angles A and B to the clifftop at the endpoints of the measured distance d, the height "h" of the cliff is given by h = d/(Cot A - Cot B) where angle A is smaller than angle B.
The sine and cosine functions can be used together to approximate more complex periodic functions. This technique, called Fourier Transforms after its inventor Joseph Fourier, can also be used to encode pictures for transmission. This is the way all signals are sent back from space. A few numbers, when plugged into a sequence of sines and cosines, can reproduce a complex picture with any degree of accuracy that you need. The more numbers in the transform, the clearer the picture.
Trigonometric functions have infinite series representations that make them useful for dealing with a wide array of mathematical functions and techniques. For example, Sin X = X - X^3/3! + X^5/5! and Cos X = 1 - X^2/2! + X^4/4! can be combined with the series representations of other functions like e^X = 1 + X/1! + X^2/2! + X^3/3! to find relationships like 2^iX = Cos X + iSin X, which is useful in a variety of mathematical applications and produces what many mathematicians consider to be one of the most amazing relationships in mathematics when X is pi: e^pi i = Cos pi - i Sin pi = -1 + i(0) = -1 or e^pi i + 1 = 0.