What is a radian?
In short, a radian is an angular measurement, very much like a degree, except one radian is a lot bigger than one degree, and it has some special properties. A radian can often seem like an unnecessarily complicated alternative to the degree measurement, and it's easy to forget why it's so useful!
While a circle is 360 degrees, a circle is also 2π radians (that is 2pi, or about 6.283). This means one radian is approximately 57.30 degrees. Not a whole number, or even a rational number! Indeed, this is one of the confusing things about radians! But this conversion is the most important thing to remember.
Two properties of the radian are particularly important;
- A simple law of circles states that the circumference equals the radius multiplied by 2pi. Because 2pi is also the number of radians in a circle, this leads to the equation for circle segments L=r*θ, where r is the radius, θ is the angle (in radians) subtended by the circle segment (the angle of the segment at the centre), and L is the arc length (the circumference of circular part of the segment).
Let's imagine a circular cake (yum!), with radius 10cm. If you cut the cake into six pieces, each piece subtends 60 degrees, or pi/3 radians, at the centre. The length of the curved edge of any slice (L) is then equal to 10*(pi/3), or about 10.47cm. If instead you cut a slightly smaller slice which subtends an angle of one radian (about 57.30 degrees), then the length of the curved edge will be exactly the same as the radius, that is, 10cm.
- If you've had to differentiate trigonometric functions, such as sin(x), you should know for example that the derivative of sin(x) is cos(x). Nice and simple, right? This is however only true when x is measured in radians.
Why is this the case? It seems almost coincidental. But by considering the graph of sin(x) this might seem more believable! If you try drawing the graph y=sin(x) with a 1:1 axis ratio, in the radian system, going through (0, 0) and (pi/2, 1), try and measure the gradient at x=0. It should give a gradient of about 1 (depending how accurate the drawing is!). Since the derivative of sin(x) is cos(x), and cos(0) = 1, you would predict the gradient at x=0 to be 1, so this makes sense!
If you tried the same with the degree system, you would need a very wide piece of paper for 1:1 axes, but it should be clear that the gradient at x=0 is much less than 1. Hopefully this makes it clear that differentiating sin(x) to cos(x) doesn't work so smoothly in the degree system.
