small angle approximation

The magnitude of an angle (in radians) is given by the fraction of the circle's circumference that it subtends. Thus a 360 degree angle subtends an arc whose length is 2 pi times the radius of the circle (this is the definition of pi - circumference divided by diameter). Dividing this by the radius gives the magnitude of the angle, 2 pi radians.  Similarly, a 180 degree angle is equivalent to pi radians, 90 degrees to pi/2 radians, and so on.

In general, the magnitude of the angle is given by the arc length subtended by the angle of a circle centered at its vertex, divided by the circle's  radius, i.e 2 pi l/2 pi r, or l/r,  where l is the arc length subtended by the angle, and r is the radius of the circle centered at the angle's vertex.  Thus, it is a dimensionless quantity.

Now consider the drawing above, with a circle whose radius is approximately equal to a, centered at V, and an angle whose sine is x/a.   For small angles, the length of the arc subtended by this angle can be approximated by x.  Hence x/a also approximates the magnitude of the angle, as defined above.

You can check this out with a table of trigonometric functions.  Up to about 5 degrees, the angle in radians and its sine are equal to within 3 or 4 decimal places.  Even at 10 degrees, the angle is 0.1745 radians, and the sine is 0.1736.  At 20 degrees, the angle is 0.3491 and the sine is 0.3420.