Express the ratios of certain sides of a right triangle as a sine or cosine. This is the simplest expression of sines and cosines. If you have a right triangle, and you are viewing the triangle from one of the smaller angles, that angle has both a sine and a cosine. The sine of the angle is the ratio of the length of the side opposite the angle divided by the hypotenuse of the triangle. The cosine is the ratio of the other short side to the hypotenuse. Tables of sines and cosines of different angles exist as printed tables and on most scientific calculators.
Draw the sine and cosine curves to get an intuitive feel for the nature of these functions. Both graphs have the same curve, but they are "out of phase" -- if you graph them together, each is the other shifted left or right. Both functions have a maximum value of plus one and a minimum value of minus one, and both functions endlessly repeat in both directions. If you start a disk rolling down the X axis with a pen attached to the edge of the disk, the pen would draw a sine or cosine curve depending on where the disk started.
Use the infinite series representations of sine and cosine to show how they relate to other mathematical functions. Sin X = X - X^3/3! + X^5/5! - X^7/7! + and so on. Cos X = 1 - X^2/2! + X^4/4! - X^6/6! + and so on. The most obvious of these relationships is to exponential functions of Euler's number: e^X = 1 + X/1! + X^2/2! + X^3/3! + and so on. This results in the very useful relationship e^iX = Cos X + i Sin X that is used to translate frames of reference in 3D geometry.