Mathematical Toolkit

Have a look at Fig. 6: If b is small compared to c, we can assume that the two longer sides of the triangle, c, have the same length as the centre line. With the usual equations for a right-angled triangle we find:
sin(β/2) = (b/2)/c
We can use the small-angle approximation sin x = x, if we are dealing with very small angles (but only when the angle is measured in radians). This approximation may seem less justified, but it can be mathematically proven to be very good for small angles.

Task AT6

Try this approximation yourself by calculating sin(1°), sin(1’), sin (1’’). Note that you have first to convert the angles from degrees to radians.

Now you have a simple relationship between b, c, and β without the trigonometric function: β/2 = (b/2)/c c = b/β


Figure 6: Dealing with small angles If b is small compared to c, this implies that β is a small angle. We can therefore get a relationship between b, c and β without trigonometric functions.

 

Units and other basic data

 

1 arcminute = 1' = 1/60 of a degree = 2.9089 × 10-4

1 arcsecond = 1'' = 1/3600 of a degree = 4.848 × 10–6 radians

1 milliarcsecond (mas) = 1/1000 arcsecond

Speed of light (c) = 2.997 × 108 m/s

1 parsec (pc) = 3.086 × 1013 km = 3.26 light-years

1 kiloparsec (kpc) = 1000 parsec

1 Megaparsec (Mpc) = 106 parsec

1 nanometer (nm) = 10–9 m