What actually are sin, cos and tan?

This is a common question asked by my students when they first learn about trigonometry. First draw a unit circle on the plane and the graphs of sine and cosine next to it. The circle obviously has 360 degrees in it. As we move around the whole circle sine takes the value of the y component on the circle. Likewise cosine takes the x component. Hence the graph varies from -1 to 1 and is periodic.

Next we define tan=sin/cos and hence we get the graph of tan, tending to infinity and minus infinity as cos tends to 0. This is a slightly more difficult question, but I use this explanation to give the student some sort of basis for sin, cos and tan rather than just using it.