T1. The slope of a line is the tangent of the angle which the line makes with the x-axis. Any of the six trigonometric funtions can be used to express all of the others. If the tangent of the angle is m, then
The reason that the tangent is not used as a basis for the other trigonometric functions is because the tangent has an infinite value for a vertical line. The sine and cosine do not have that problem, so they are used more commonly. For our purposes, the tangent will crop up more commonly.
In this theorem, note that the formula for the secant appears. That is because the slope distance is the horizontal distance times the secant which we see.
T2. The perpendicular distance between the two lines is the distance between the y intercepts times the sine of the angle which the line makes with the y axis which is the cosine of the angle it makes with the x axis.