Given the functions (sina, cosa, sinb and cos b), we seek formulas that express sin(a+b) and cos(a+b). The first of these formulas is used in deriving the L4 and L5 Lagrangian points, here.
Please verify every calculation step before proceeding!
ABD which " " " b
AC = R cos b
AB = AC cos a = R cos b cos a
R cos (a+b) = AF Start by deriving the sine:
In the rightangled triangle CED
EC = DC sin a = R sin b sin a
AB = R cos b cos a R sin (a+b) = BC+DE = R cos b sin a + R sin b cos a Cancelling R and rearranging a to precede b sin (a+b) = cos b sin a + sin b cos a
Similarly, for the cosineR cos (a+b) = AF = AB  FB = AB  EC = = R cos b cos a  R sin b sin a Cancelling R and rearranging cos (a+b) = cos a cos b  R sin a sin b

#34b The L4 and L5 Lagrangian Points
Author and curator: David P. Stern, u5dps@lepvax.gsfc.nasa.gov
Last updated 25 February 1999