If I take my transmission, and I rotate it at a constant speed, lets say 10 rpms, then the output shaft on the transmission will also spin at 10 rpms. Now lets introduce a driveshaft into the equation and connect it to our output shaft with a u-joint. We will angle this shaft downwards a few degrees so that the explanation of this will make sense. Remember from before when I said that as the angle increases, the more the u-joint will cause the connected shaft to speed up and down? Okay, lets put that to use. Since our transmission is spinning at a constant rate, and the driveshaft we have now installed is at an angle, the driveshaft is now spinning, but its speed is increasing, then decreasing, then increasing, and so on. See the graph below that demonstrates this speed increase and decrease. The zero line represents a constant speed. When the curve goes above the zero line, the driveshaft speed increases; and when it goes below the zero line the speed decreases.

Image from wikipedia.org

Now lets connect an axle to our system. But instead of using a u-joint, we just weld the driveshaft onto the pinion. This now means that as we drive down the road, the rear tires will constantly be speeding up, then slowing down, then speeding up, then slowing down, etc. Are you with me so far?

Lets get rid of those welds between the axle and the driveshaft, and introduce a u-joint in the picture, just as we did between the transmission and driveshaft. We want to cancel out this speed increase and decrease, and the way we do that is by setting the angle between the driveshaft and pinion to the same as the angle between the driveshaft and transmission.

Let me interject something for a moment. Some people say "Well my transmission points down 3°, and my driveshaft points up 2°, and my pinion is up 5°." Well that's all fine and dandy, but it doesn't really mean much and it won't help you in setting your angles properly. We don't want to simply look at the angles of these objects in reference to Mother Earth. We need to look at them in relationship to each other. In this case, the angle between the transmission and driveshaft is 5° (3+2), and the angle between the driveshaft and pinion is 7° (5+2). These 5° and 7° angles are called operating angles.

Ok, so back to canceling this speeding up and slowing down rotation. Using the simple example above, we can see that the angle between transmission output shaft and the driveshaft is 5°. In order to cancel the angular difference, the angle between the driveshaft and pinion needs to be 5°. Now, we can go two ways with this. The pinion can either point up 3°, or it can point down 3°. Why? Simple. If it points up 3°, that means the difference between the two is 5° (5+3). If it points down 3°, then again, 3+2 = 5°. The object is to get the same angle as the output shaft. There are no 'negative' angles, persay. This means that if the first angle is 5°, the second angle 5°, one way or the other. See the two illustrations below:

Image from http://www.roddingroundtable.com/

In simple terms, if your transmission output shaft is pointing either up or down at any specific angle then your pinion should point at the same angle. If the transmission is pointing down towards the rear end at some angle, lets say 3°, then your pinion should point up towards the front of the vehicle at 3°. But what happens when we introduce a double cardan joint into the system. Read on...