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(M-11) Deriving sin(a+b), cos(a+b)
Take Note! The illustration below uses the Greek letters ""alpha" and "beta", written as a and b. If all you see here are the letters "a" and "b", your browser is overriding the font specification, and everywhere below (in the title above, too), these letters appear as "a" and "b."
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!
As shown in the drawing, to derive the formula we combine two right-angled triangles
ABC which has an angle a
The long side ("hypotenuse') of ACD is AD=R. Therefore
ACD which " " " b
DC = R sin b
AC = R cos b
BC = AC sin a = R cos b sin a
The triangle ADF is right-angled and has the angle (a+b). Therefore
AB = AC cos a = R cos b cos a
R sin (a+b) = DF
R cos (a+b) = AF
Start by deriving the sine:
R sin (a+b) = DF = EF + DE = BC + DE
Note in the drawing the two head-to-head angles marked with double lines: like all such angles, they must be equal. Each of them is one of the two sharp ("acute") angles in its own right-angled triangle. Since the sharp angles in such a triangle add up to 90 degrees, the other two sharp angles must be equal. This justifies marking the angle near D as a, as drawn in the figure.
In the right-angled triangle CED
DE = DC cos a = R sin b cos a
Earlier it was already shown that
EC = DC sin a = R sin b sin a
BC = R cos 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 cosine
R 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 - sin a sin b
Application of these formulas: #34b The L4 and L5 Lagrangian Points
Trigonometry Proficiency Drill
More "Trig": The Tangent
Author and Curator: Dr. David P. Stern
Mail to Dr.Stern: audavstern("at" symbol)erols.com .
Last updated 25 November 2001
Above is background material for archival reference only.