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Please verify every calculation step before proceeding!
ACD which " " " β
AC = R cos β
AB = AC cos α = R cos β cos α
R cos (α+β) = AF Start by deriving the sine:
In the rightangled triangle CED
EC = DC sin α = R sin β sin α
AB = R cos β cos α R sin (α+β) = BC+DE = R cos β sin α + R sin β cos α Cancelling R and rearranging α to precede β sin (α+β) = cos β sin α + sin β cos α
Similarly, for the cosineR cos (α+β) = AF = AB –FB = AB –EC = = R cos β cos α – R sin β sin α Cancelling R and rearranging cos (α+β) = cos α cos β – sin α sin β

Application of these formulas: #34b The L4 and L5 Lagrangian Points M11a. Trigonometry Proficiency Drill More "Trig": M12. The Tangent
Author and Curator: Dr. David P. Stern 
Curators: Robert Candey, Alex Young, Tamara Kovalick
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