Why do we use radians? After all, there’s nothing wrong with degrees. Everyone understands degrees. Almost any fraction of a circle is a whole number of degrees. In radians, it seems like every angle is an irrational number. So what’s the point?
The truth is that no-one uses radians to measure angles. We use radians because it makes the graph of sin(x) look nice. If you draw the graph of sin(x) in degrees, it is a virtually flat, featureless graph: the y-axis values vary only from –1 to +1, yet the x-axis values vary from 0° to 360°. But if you draw the graph of sin(x) in radians, the x-axis values vary from 0 to about 6. So the graph has the familiar S-shape.
This is not a trivial point. Look at the graph of sin(x) in radians near to the origin. It looks like a straight line. Specificially, it looks like the line y = x. So, for small values of x, sin(x) is pretty much the same as x – the sine of an angle is equal (almost) to the angle itself, provided the angle is small and is measured in radians.
So what is sin(18°)? Well, 18° is a tenth of 180°, so it’s a tenth of π, i.e. about 0.31. So the sine of 18° is equal to the sine of 0.31 radian. But the sine of angle in radians is approximately equal to the angle itself. Thus sin(18°) ≈ sin(0.31) ≈ 0.31. Easy!
(If you check this on a calculator, you’ll see it’s correct!)
But how do we find sin(54°) – for angles that large, the sine graph doesn’t look at all like a straight line. So what do we do then? What does the calculator do? (Hint: the graph of sin(x) looks a little bit like a cubic equation between –180° and +180°.)