The discussion focuses on finding the minimum value of the product tan(a)·tan(b)·tan(c) for acute angles a, b, and c, where a + b + c = 180°. Participants suggest using the relationship between the tangents of the angles and the geometric and arithmetic means to solve the problem. Additionally, the second question involves proving that tan(a) + tan(b) + tan(c) ≥ √3 when a + b + c = 90°, with hints to utilize the Lagrange multiplier method for optimization. The conversation emphasizes the importance of demonstrating initial attempts to solve the problems.
PREREQUISITESStudents studying precalculus mathematics, particularly those interested in trigonometry and optimization techniques, as well as educators seeking to enhance their teaching methods in these topics.
NeroKid said:you can use the langrange function to do it, it is quite easy if u use it
Michael_Light said:1. Given a,b,c are acute angles and a + b + c =180. find the minimum of tan(a).tan(b).tan(c)
ehild said:Following rock.freak's suggestion, I would write the tangent of the third angle in terms of the tangents of the other two angles, and would use the relation between geometric and arithmetic means.
ehild
NeroKid said:tan(a)*tan(b)*tan(c) = -tan(b+c) *tan(b)*tan(c) = (tanb+tanc)/(1-tanb*tanc) *tanb*tanc
call S = tanb + tanc and P = tanb * tanc S^2>= 4P this should be easy from here on
Michael_Light said:Still cannot do...