The discussion revolves around the derivation of a triangle approximation using the law of cosines. Participants are examining the relationship between the sides of a triangle and the angles involved, particularly in the context of small angle approximations.
The discussion is ongoing, with participants sharing their attempts and questioning specific factors in the derivation, such as the presence of the sin²(B) term. Some guidance on using Taylor series has been suggested, but no consensus has been reached.
There is a reference to a drawing that contains necessary variables, but the specifics of the drawing are not included in the discussion. Participants are also grappling with the implications of small angle approximations and the relationships between the triangle's sides and angles.
GroupTheory1 said:The law of cosines says that
b^2 - a^2 = c^2 - 2ac * cos(B),
so,
b - a = -2ac * cos(B) / (b+a) + c^2 / (b+a).
If one says that a ~ b because c << a, then the above becomes
b - a ~ -c * cos(B) + c^2 / (2a),
which is close but does not match and is way off for B approaching zero degrees.
Sourabh N said:As I read the figure and understand your notation, B = [tex]\pi[/tex] - ([tex]\theta[/tex] + [tex]\gamma[/tex]), thus what you got equals what's on the figure.