Working out sin18° in your head is easy!

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, 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 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) 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 measured in radians. (If you click on the graph above you will get a much larger version.) 

So what is sin18°? Well, 18° is a tenth of 180°, so it's a tenth of pi, 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 sin18° is about 0.31.

Easy!

But how do we find sin54° -- for angles that large, the sine graph doesn't look at all like a straight line. So what do we do? What does the calculator do?