Who Cares about 53.13 Degrees?

You just calculated the following values for a vehicle with a heading of 53.13 degrees

d " style="text-align:center;"> D	\delta_y " style="text-align:center;"> \Delta y	\delta_x " style="text-align:center;"> \Delta x
5	4	3
10	8	6
1	4/5	3/5

But really, for this particular angle we can say something much more general . We can say:

\Delta y = \frac{4}{5}\times D

\Delta x = \frac{3}{5}\times D

And these are very useful equations! They tell us that the vertical displacement is equal to total displacement times some multiplier (in this case ⅘). Likewise, horizontal displacement is total displacement times some other multiplier (in this case ⅗).

And this is useful…. for all those times when you're driving at 53.13 degrees?

What about every other possible angle ? Wouldn't it be nice to calculate these multipliers for any angle?