The most obvious measure of direction is an angle. This program can draw a line and a circle that are tangent, at any angle. You can see the angle, in degrees, in the program.
For general curves that can go in any direction, such as a circle or any other loop, this is often the most appropriate measure.
But for continuous, single-valued functions, which cannot turn backwards, there is another measure that leads us to many useful consequences.
There is no tangent at a point of discontinuity.
A continuous, multiple-valued function necessarily turns backward in some interval.
The tangent to a continuous, single-valued function has an angle between -90 degrees and 90 degrees.
If we take any angle inside the range [-90,90], and take the x and y coordinates of a point on one side, we can build a right triangle with that angle and with sides equal to the coordinates, thus.
If we make the lower side of a fixed length, and take that length as the unit of measure, the length of the other side is a function of the angle. In the following diagram, we use a radius of a triangle as the fixed side. The other side in this triangle is tangent to the circle.
This function of the angle is called the tangent function.
The tangent function gives us a convenient measure of the direction of a tangent line, one that will lead us into calculus. This is often called the slope of the function. We have seen the tangent function defined using a fixed radius, but it works for any x value. Going back to our diagram of an angle, a point and its coordinates, and the triangle built from those coordinates, the slope, or the tangent of the angle, is the y coordinate divided by the x coordinate, y/x. Since y is how much the line rises when we run from 0 to x, the tangent can be described as rise/run (rise over run). Some students find this a helpful mnemonic, while others find it completely confusing.
The next steps in the use of slopes are the derivative function, the definite integral, and the Fundamental Theorem of the Calculus, which will be the subjects of further tutorials.