The trig functions sine, cosine, and tangent all have interpretations in the unit circle. The unit circle is a circle of radius 1 with its center at the origin. Imagine a point P on the circle. Now imagine the ray* R drawn from the center of the circle (ie, the origin of the plane) through point P and out to infinity. This ray forms an angle with the positive x-axis. This angle is measured as the counterclockwise 'sweep' from the x-axis to the ray R. Call this angle theta. Now, we say that the sine of theta is equal to the y coordinate of the point. The cosine of theta is the x coordinate of the point. Got it so far? The tangent requires a slight addition to our diagram. Imagine a line T parallel to the y-axis and that touches the unit circle at the point where the circle crosses the x-axis. That is, the line T is tangent to the circle at this point on the x-axis. Note that for most angles theta the ray R crosses the line T at a point. So finally, we can say that the tangent of theta is the y coordinate of this new point!** ----- * A ray is different from a line in that a ray has one endpoint. A line goes to infinity in both directions, and a segment has two endpoints. ** For what angles theta will the tangent not exist?

The tangent (sin/cos) of an angle is the slope (rise/run) of a line drawn at that angle to a horizontal axis. The slope of the line tangent to the circle (see unit circle information in another post) at the point where a line at that angle from the center intersects the circle is -1/tan, i.e. perpendicular to the original line. So, the two concepts are related in a way. I confess ignorance as to why they were both given the same name... maybe it is an ancient Greek thing...??? Hopefully somebody else knows.

A tangent to a curve, at a point on that curve, on an xy graph, is the line that passes through the point and has the same slope as the curve does at that point. This meaning of tangent is more formally defined in introductory calculus. The slope of the line is how steep it is: how fast it rises in y for each unit of x. This can be specified as a ratio, e.g. "a one in five slope", or a percentage": a twenty percent slope", or a fraction: "one fifth". The tangent function gives the slope of the tangent line in terms of the anticlockwise angle from the horizontal (specifically the positive x direction). So there's a direct relationship between the tangent to a curve and the tangent function. Finally the tangent to the circle is just a specific example of a tangent line.

Insofar as the slope of the hypotenuse of a right triangle is sometimes called rise/run, the tangent of the angle in the unit circle is the same. Sin/cos (or opposite over adjacent) is the slope of the hypotenuse. This has a practical use when finding a tangent to a curve at a given point. Try this: take a curve, make two points on the curve, draw a triangle with sides parallel to the x and y axis. See that the slope of the line is the tangent of the angle formed in the triangle. As you move the points closer together, the triangle gets smaller, but the tangent gets closer to the true slope of the line through a point.

This diagram should explain the geometric origin of the function name (involves similar triangles): http://en.wikipedia.org/wiki/Image:Circle-trig6.svg

NO.