In any circle there are 360 degrees, written as 3600. This defines a basic angle, the "degree", for measuring any other angle against: one degree is that small angle of which there are 360 in a circle.
So when we say "an angle of 900", we mean an angle which is 90 times as big as our basic angle, the degree.
An alternative angle to use as a basic unit to measure other angles against is a radian. This basic angle is about 57 times as big as a degree, as shown below.
Naturally, since a radian is so much bigger than a degree, there are fewer radians in a circle. In fact there's only about 6.
There is a complication with using the radian as your basic angle instead of the degree, but that complication is also the reason why using the radian is often more useful.
The complication is this: there isn't a whole number of radians in a circle. I said there are about 6 of them, in fact it's more like 6.28. In fact, it's more like 6.28319 of them. In fact, ...
Actually this could go on forever, not only is there not a whole number of radians in a circle, but the number of radians in a circle is 2p , where p is the Greek letter "pi" and represents a so-called "irrational" number whose value is about 3.14.
Again I can't tell you its value exactly because its decimal version goes on forever and there's no way of writing it as a fraction either (that's what we mean by "irrational").
It seems like an inconvenient number to deal with, but the number p is fundamental to any circle. That's because of the fact that:
the circumference, C, of any circle of radius r is given by 2pr.
So for any circle, no matter how big or small, the circumference is always 2p times as long as the radius.
This fact gives us the reason for using radians: if there are 2p radius's in a circumference, and there's 2p radians in a circle, then the length of circumference covered by going through an angle of one radian must be one radius!
This is illustrated below: here the radius is 5 and so the part of the circumference covered by going through an angle of one radian is also 5.
It's actually not so difficult to work with the number p, we just refer to it as p whenever it crops up and very rarely actually need to put in its (somewhat inconvenient) numerical value.
So if we use radians, instead of thinking that going round a circle is going through 360 degrees, so that the angle in half a circle is 180 degrees and that in a quarter-circle (i.e. a right angle) is therefore 90 degrees, we just start with the angle in a full circle being 2p, so the angle in a semi-circle is p and a right-angle is p/2.
We can turn any angle that we know in degrees into an angle in radians by dividing by 360 and multiplying by 2p.
Here's a picture to show some of the most commonly-used angles interms of degrees and radians.
We'll see in a later section that these are "special" angles because their "sine" and "cosine" values can be written down exactly from what we know about the sides of triangles containing these angles. It's useful to remember these angles in both degrees and radians.