Slopes of lines!

One reason that tangents are so important is that they give the slopes of straight lines. Consider the straight line drawn in the x-y coordinate plane.

The point B is where the line cuts the y-axis. We can let the coordinates of B be (0,b) so that b, called the y-intercept, indicates how far above the x-axis B lies. (This notation conflicts with labeling the sides of a triangle a, b, and c, so let's not label the sides right now.)
You can see that the point 1 unit to the right of the origin is labeled 1, and its coordinates, of course, are (1,0). Let C be the point where that verical line cuts the horizontal line through B. Then C has coordinates (1,b).
The point A is where the vertical line above 1 cuts the original line. Let m denote the distance that A is above C. Then A has coordinates (1,b+m). This value m is called the slope of the line. If you move right one unit anywhere along the line, then you'll move up m units.
Now consider the angle CBA. Let's call it the angle of slope. It's tangent is CA/BC = m/1 = m. Therefore, the slope is the tangent of the angle of slope.