Introduction
When we were introduced to calculus during 'A' levels (actually 'AO' levels), an investigation into differentiation was used to show that sin(a)≈a for small angles, if the angles were in radians.
At the time, the observation seemed fairly magical: that somehow by picking radians as the units for degrees made it work out. To me, radians always seemed an arbitrary unit, because it wasn't a whole number, but 2π for a complete circle. And the method didn't help to enlighten us, because it involved repeated calculations for sin(a+d)-sin(a) for a⟶0.
Actually, though the reason is very simple. Let's consider a full circle with a unit radius:
The distance around the circle (the circumference) is 2π and any part of that distance a around the circle is its angle in radians (since 2π = 360º). Now let's look at a right-angled triangle embedded in a small part of the arc:
The height of the triangle is h, is nearly the same as the length a of the arc around the circle for angle x. And the height of the triangle is the actual meaning of sin(a), since the length a of the arc is the same as the angle in radians, because the radius=1. If we just enlarge that part, we can see that it's very close, but not quite the same.
