On an x-y plane as shown to the right, we define the trigonometric ratios in terms of x, y and r:

Notice that we are still defining:

 

However, we are using the specific x-, y- and r-values defined by the point (x, y). We can choose any point on that line, of course, to define our ratios.

Not surprisingly, the reciprocal ratios are defined similarly in terms of the x-, y- and r-values as follows:

 

Double angle & half angle formula